{"id":401,"date":"2024-04-18T17:42:23","date_gmt":"2024-04-18T17:42:23","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=401"},"modified":"2025-08-20T16:00:12","modified_gmt":"2025-08-20T16:00:12","slug":"algebra-essentials-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/algebra-essentials-background-youll-need-1\/","title":{"raw":"Algebra Essentials: Background You'll Need 1","rendered":"Algebra Essentials: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Use prime factorization techniques to break down numbers into their prime factors.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>What are Prime and Composite Numbers?<\/h2>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>prime numbers<\/h3>\r\n<strong>Prime Numbers<\/strong> are natural numbers greater than [latex]1[\/latex] that have only two distinct positive divisors: [latex]1[\/latex] and themselves.\r\n\r\n<\/section>This means they can only be divided evenly (without leaving a remainder) by 1 and the number itself. Prime numbers are the building blocks of all natural numbers because every number can be expressed as a product of primes.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\"><strong>Examples of Prime Numbers: <\/strong>[latex]2[\/latex] (note: it is the only even prime number), [latex]3, 5, 7, 11, 29[\/latex].<\/section><section class=\"textbox keyTakeaway\">\r\n<h3>composite numbers<\/h3>\r\n<strong>Composite Numbers<\/strong> are natural numbers that have more than two positive divisors.\r\n\r\n<\/section>In other words, aside from being divisible by [latex]1[\/latex] and the number itself, composite numbers can be divided evenly by at least one other number.\r\n\r\n<section class=\"textbox example\" aria-label=\"Example\"><strong><strong>Examples of Composite Numbers: <\/strong><\/strong>[latex]4[\/latex] (divisors: [latex]1, 2, 4[\/latex]) and [latex]15[\/latex] (divisors: [latex]1, 3, 5, 15[\/latex]).<\/section><section class=\"textbox proTip\">The number [latex]1[\/latex] is neither prime nor composite! It is unique in that it has only one positive divisor (itself).<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18648[\/ohm2_question]<\/section>\r\n<h2 class=\"title\">Prime Factorization<\/h2>\r\nThe prime factorization of a number is the product of prime numbers that equals the number. You may want to refer to the following list of prime numbers less than [latex]50[\/latex] as you work through this section.\r\n<p style=\"text-align: center;\">[latex]2,3,5,7,11,13,17,19,23,29,31,37,41,43,47[\/latex]<\/p>\r\n\r\n<section class=\"textbox proTip\">Memorizing the first five prime numbers ([latex]2, 3, 5, 7, 11[\/latex]) will come in handy when reducing fractions.<\/section>\r\n<h3>Prime Factorization Using the Factor Tree Method<\/h3>\r\nOne way to find the prime factorization of a number is to make a factor tree. We start by writing the number, and then writing it as the product of two factors. We write the factors below the number and connect them to the number with a small line segment\u2014a \"branch\" of the factor tree. If a factor is prime, we circle it (like a bud on a tree), and do not factor that \"branch\" any further. If a factor is not prime, we repeat this process, writing it as the product of two factors and adding new branches to the tree. We continue until all the branches end with a prime. When the factor tree is complete, the circled primes give us the prime factorization. For example, let\u2019s find the prime factorization of [latex]36[\/latex]. We can start with any factor pair such as [latex]3[\/latex] and [latex]12.[\/latex] We write [latex]3[\/latex] and [latex]12[\/latex] below [latex]36[\/latex] with branches connecting them.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"189\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220021\/CNX_BMath_Figure_02_05_018_img.png\" alt=\"A factor tree with the number 36 at the top. Two branches are splitting out from under 36. The right branch has a number 3 at the end with a circle around it. The left branch has the number 12 at the end.\" width=\"189\" height=\"102\" \/> Factor Tree of 36 (Partial)[\/caption]\r\n\r\n&nbsp;\r\n\r\nThe factor [latex]3[\/latex] is prime, so we circle it. The factor [latex]12[\/latex] is composite, so we need to find its factors. Let\u2019s use [latex]3[\/latex] and [latex]4[\/latex]. We write these factors on the tree under the [latex]12[\/latex].\r\n\r\n&nbsp;\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"180\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220022\/CNX_BMath_Figure_02_05_019_img.png\" alt=\"A factor tree with the number 36 at the top. Two branches are splitting out from under 36. The right branch has a number 3 at the end with a circle around it. The left branch has the number 12 at the end. Two more branches are splitting out from under 12. The right branch has the number 4 at the end and the left branch has the number 3 at the end.\" width=\"180\" height=\"146\" \/> Factor Tree of 36 (Expanded)[\/caption]\r\n\r\n&nbsp;\r\n\r\nThe factor [latex]3[\/latex] is prime, so we circle it. The factor [latex]4[\/latex] is composite, and it factors into [latex]2\\cdot 2[\/latex]. We write these factors under the [latex]4[\/latex]. Since [latex]2[\/latex] is prime, we circle both [latex]2\\text{s}[\/latex].\r\n\r\n&nbsp;\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"185\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220023\/CNX_BMath_Figure_02_05_009_img.png\" alt=\"A factor tree with the number 36 at the top. Two branches are splitting out from under 36. The right branch has a number 3 at the end with a circle around it. The left branch has the number 12 at the end. Two more branches are splitting out from under 12. The right branch has the number 4 at the end and the left branch has the number 3 at the end with a circle around it. Two more branches are splitting out from under 4. Both the left and right branch have the number 2 at the end with a circle around it.\" width=\"185\" height=\"211\" \/> Complete Factor Tree of 36[\/caption]\r\n\r\n&nbsp;\r\n\r\nThe prime factorization is the product of the circled primes. We generally write the prime factorization in order from least to greatest.\r\n<p style=\"text-align: center;\">[latex]2\\cdot 2\\cdot 3\\cdot 3[\/latex]<\/p>\r\nIn cases like this, where some of the prime factors are repeated, we can write prime factorization in exponential form.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}2\\cdot 2\\cdot 3\\cdot 3\\\\ \\\\ {2}^{2}\\cdot {3}^{2}\\end{array}[\/latex]<\/p>\r\nNote that we could have started our factor tree with any factor pair of [latex]36[\/latex]. We chose [latex]12[\/latex] and [latex]3[\/latex], but the same result would have been the same if we had started with [latex]2[\/latex] and [latex]18,4[\/latex] and [latex]9[\/latex], or [latex]6[\/latex] and [latex]6[\/latex].\r\n\r\n<section class=\"textbox questionHelp\">\r\n<p class=\"title\"><strong>How to: Find the prime factorization of a composite number using the tree method<\/strong><\/p>\r\n\r\n<ol id=\"eip-id1168469875559\" class=\"stepwise\">\r\n \t<li>Find any factor pair of the given number, and use these numbers to create two branches.<\/li>\r\n \t<li>If a factor is prime, that branch is complete. Circle the prime.<\/li>\r\n \t<li>If a factor is not prime, write it as the product of a factor pair and continue the process.<\/li>\r\n \t<li>Write the composite number as the product of all the circled primes.<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\">Let's take a look at the number [latex]48.[\/latex] Note: We can also visualize the decomposition of a composite number into its prime components, much like creating a blueprint for a building but with numbers, using a factor tree. Each 'branch' of our factor tree represents a division by a prime factor.\r\n<table id=\"eip-id1168466026521\" class=\"unnumbered unstyled\" style=\"width: 100%;\" summary=\"The figure shows multiple factor trees with the number 48 at the top. In the first tree two branches are splitting out from under 48. The branches use the factor pair 2 and 24 with 24 at the end of the right branch and 2 at the end of the left branch. Two has a circle around it to show that it is prime and that branch is complete. In the next tree the previous tree is repeated, but now with two branches splitting out from under 24. The branches use the factor pair 4 and 6 with 6 at the end of the right branch and 4 at the end of the left branch. Neither of these factors is circled because they are not prime. In the last tree the previous tree is repeated, but now with two branches splitting out from under 4 and two branches splitting out from under 6. The branches under 4 use the factor pair 2 and 2. Both of these two's are circled to show that they are prime and that branch is complete. The branches under 6 use the factor pair 2 and 3. Both of these numbers are circled to show that they are prime and that branch is complete. The prime factorization of the number 48 is made up of all of the circled numbers from the factor tree which is 2, 2, 2, 2, and 3. The prime factorization can be written as 2 times 2 times 2 times 2 times 3 or using exponents for repeated multiplication of 2 it can be written as 2 to the fourth power times 3.\">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 70.14%;\">Ask yourself, 'What prime number is a factor of [latex]48[\/latex]?'\r\n<ul>\r\n \t<li>Well, the smallest prime number is [latex]2[\/latex].<\/li>\r\n \t<li>Divide [latex]48[\/latex] by [latex]2[\/latex], giving you [latex]24[\/latex].<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td style=\"width: 29.0824%;\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"293\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220024\/CNX_BMath_Figure_02_05_022_img-01.png\" alt=\".\" width=\"293\" height=\"108\" \/> Factor Tree of 48(Partial)[\/caption]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 70.14%;\">Next, consider [latex]24[\/latex].\r\n<ul>\r\n \t<li>It's not a prime number, so what two numbers multiply to get [latex] 24[\/latex]?<\/li>\r\n \t<li>One option is [latex]4[\/latex] and [latex]6[\/latex].<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td style=\"width: 29.0824%;\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"293\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220025\/CNX_BMath_Figure_02_05_022_img-02.png\" alt=\".\" width=\"293\" height=\"181\" \/> Factor Tree of 48 (Expanded)[\/caption]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 70.14%;\">\r\n<ul>\r\n \t<li>Notice that [latex]4[\/latex] is not a prime number, but can be broken down further into [latex]2[\/latex] and [latex]2[\/latex].<\/li>\r\n \t<li>[latex]6[\/latex] can also be decomposed. Break it down into [latex]2[\/latex] and [latex]3[\/latex], with 3 being a prime number.<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td style=\"width: 29.0824%;\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"293\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220026\/CNX_BMath_Figure_02_05_022_img-03.png\" alt=\".\" width=\"293\" height=\"266\" \/> Factor Tree of 48 (Complete)[\/caption]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 70.14%;\">We have fully factored down [latex]48[\/latex] into its prime factors.\r\nLet's write the product of all of the prime factors from least to greatest.<\/td>\r\n<td style=\"width: 29.0824%;\">[latex]48 = 2\\cdot 2\\cdot 2\\cdot 2\\cdot 3[\/latex]\r\n\r\nIn [pb_glossary id=\"756\"]exponential form[\/pb_glossary]: [latex]48 = 2^4 \\cdot 3[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18649[\/ohm2_question]<\/section><section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]18650[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Use prime factorization techniques to break down numbers into their prime factors.<\/li>\n<\/ul>\n<\/section>\n<h2>What are Prime and Composite Numbers?<\/h2>\n<section class=\"textbox keyTakeaway\">\n<h3>prime numbers<\/h3>\n<p><strong>Prime Numbers<\/strong> are natural numbers greater than [latex]1[\/latex] that have only two distinct positive divisors: [latex]1[\/latex] and themselves.<\/p>\n<\/section>\n<p>This means they can only be divided evenly (without leaving a remainder) by 1 and the number itself. Prime numbers are the building blocks of all natural numbers because every number can be expressed as a product of primes.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\"><strong>Examples of Prime Numbers: <\/strong>[latex]2[\/latex] (note: it is the only even prime number), [latex]3, 5, 7, 11, 29[\/latex].<\/section>\n<section class=\"textbox keyTakeaway\">\n<h3>composite numbers<\/h3>\n<p><strong>Composite Numbers<\/strong> are natural numbers that have more than two positive divisors.<\/p>\n<\/section>\n<p>In other words, aside from being divisible by [latex]1[\/latex] and the number itself, composite numbers can be divided evenly by at least one other number.<\/p>\n<section class=\"textbox example\" aria-label=\"Example\"><strong><strong>Examples of Composite Numbers: <\/strong><\/strong>[latex]4[\/latex] (divisors: [latex]1, 2, 4[\/latex]) and [latex]15[\/latex] (divisors: [latex]1, 3, 5, 15[\/latex]).<\/section>\n<section class=\"textbox proTip\">The number [latex]1[\/latex] is neither prime nor composite! It is unique in that it has only one positive divisor (itself).<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm18648\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=18648&theme=lumen&iframe_resize_id=ohm18648&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h2 class=\"title\">Prime Factorization<\/h2>\n<p>The prime factorization of a number is the product of prime numbers that equals the number. You may want to refer to the following list of prime numbers less than [latex]50[\/latex] as you work through this section.<\/p>\n<p style=\"text-align: center;\">[latex]2,3,5,7,11,13,17,19,23,29,31,37,41,43,47[\/latex]<\/p>\n<section class=\"textbox proTip\">Memorizing the first five prime numbers ([latex]2, 3, 5, 7, 11[\/latex]) will come in handy when reducing fractions.<\/section>\n<h3>Prime Factorization Using the Factor Tree Method<\/h3>\n<p>One way to find the prime factorization of a number is to make a factor tree. We start by writing the number, and then writing it as the product of two factors. We write the factors below the number and connect them to the number with a small line segment\u2014a &#8220;branch&#8221; of the factor tree. If a factor is prime, we circle it (like a bud on a tree), and do not factor that &#8220;branch&#8221; any further. If a factor is not prime, we repeat this process, writing it as the product of two factors and adding new branches to the tree. We continue until all the branches end with a prime. When the factor tree is complete, the circled primes give us the prime factorization. For example, let\u2019s find the prime factorization of [latex]36[\/latex]. We can start with any factor pair such as [latex]3[\/latex] and [latex]12.[\/latex] We write [latex]3[\/latex] and [latex]12[\/latex] below [latex]36[\/latex] with branches connecting them.<\/p>\n<p>&nbsp;<\/p>\n<figure style=\"width: 189px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220021\/CNX_BMath_Figure_02_05_018_img.png\" alt=\"A factor tree with the number 36 at the top. Two branches are splitting out from under 36. The right branch has a number 3 at the end with a circle around it. The left branch has the number 12 at the end.\" width=\"189\" height=\"102\" \/><figcaption class=\"wp-caption-text\">Factor Tree of 36 (Partial)<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<p>The factor [latex]3[\/latex] is prime, so we circle it. The factor [latex]12[\/latex] is composite, so we need to find its factors. Let\u2019s use [latex]3[\/latex] and [latex]4[\/latex]. We write these factors on the tree under the [latex]12[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<figure style=\"width: 180px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220022\/CNX_BMath_Figure_02_05_019_img.png\" alt=\"A factor tree with the number 36 at the top. Two branches are splitting out from under 36. The right branch has a number 3 at the end with a circle around it. The left branch has the number 12 at the end. Two more branches are splitting out from under 12. The right branch has the number 4 at the end and the left branch has the number 3 at the end.\" width=\"180\" height=\"146\" \/><figcaption class=\"wp-caption-text\">Factor Tree of 36 (Expanded)<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<p>The factor [latex]3[\/latex] is prime, so we circle it. The factor [latex]4[\/latex] is composite, and it factors into [latex]2\\cdot 2[\/latex]. We write these factors under the [latex]4[\/latex]. Since [latex]2[\/latex] is prime, we circle both [latex]2\\text{s}[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<figure style=\"width: 185px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220023\/CNX_BMath_Figure_02_05_009_img.png\" alt=\"A factor tree with the number 36 at the top. Two branches are splitting out from under 36. The right branch has a number 3 at the end with a circle around it. The left branch has the number 12 at the end. Two more branches are splitting out from under 12. The right branch has the number 4 at the end and the left branch has the number 3 at the end with a circle around it. Two more branches are splitting out from under 4. Both the left and right branch have the number 2 at the end with a circle around it.\" width=\"185\" height=\"211\" \/><figcaption class=\"wp-caption-text\">Complete Factor Tree of 36<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<p>The prime factorization is the product of the circled primes. We generally write the prime factorization in order from least to greatest.<\/p>\n<p style=\"text-align: center;\">[latex]2\\cdot 2\\cdot 3\\cdot 3[\/latex]<\/p>\n<p>In cases like this, where some of the prime factors are repeated, we can write prime factorization in exponential form.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{c}2\\cdot 2\\cdot 3\\cdot 3\\\\ \\\\ {2}^{2}\\cdot {3}^{2}\\end{array}[\/latex]<\/p>\n<p>Note that we could have started our factor tree with any factor pair of [latex]36[\/latex]. We chose [latex]12[\/latex] and [latex]3[\/latex], but the same result would have been the same if we had started with [latex]2[\/latex] and [latex]18,4[\/latex] and [latex]9[\/latex], or [latex]6[\/latex] and [latex]6[\/latex].<\/p>\n<section class=\"textbox questionHelp\">\n<p class=\"title\"><strong>How to: Find the prime factorization of a composite number using the tree method<\/strong><\/p>\n<ol id=\"eip-id1168469875559\" class=\"stepwise\">\n<li>Find any factor pair of the given number, and use these numbers to create two branches.<\/li>\n<li>If a factor is prime, that branch is complete. Circle the prime.<\/li>\n<li>If a factor is not prime, write it as the product of a factor pair and continue the process.<\/li>\n<li>Write the composite number as the product of all the circled primes.<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\">Let&#8217;s take a look at the number [latex]48.[\/latex] Note: We can also visualize the decomposition of a composite number into its prime components, much like creating a blueprint for a building but with numbers, using a factor tree. Each &#8216;branch&#8217; of our factor tree represents a division by a prime factor.<\/p>\n<table id=\"eip-id1168466026521\" class=\"unnumbered unstyled\" style=\"width: 100%;\" summary=\"The figure shows multiple factor trees with the number 48 at the top. In the first tree two branches are splitting out from under 48. The branches use the factor pair 2 and 24 with 24 at the end of the right branch and 2 at the end of the left branch. Two has a circle around it to show that it is prime and that branch is complete. In the next tree the previous tree is repeated, but now with two branches splitting out from under 24. The branches use the factor pair 4 and 6 with 6 at the end of the right branch and 4 at the end of the left branch. Neither of these factors is circled because they are not prime. In the last tree the previous tree is repeated, but now with two branches splitting out from under 4 and two branches splitting out from under 6. The branches under 4 use the factor pair 2 and 2. Both of these two's are circled to show that they are prime and that branch is complete. The branches under 6 use the factor pair 2 and 3. Both of these numbers are circled to show that they are prime and that branch is complete. The prime factorization of the number 48 is made up of all of the circled numbers from the factor tree which is 2, 2, 2, 2, and 3. The prime factorization can be written as 2 times 2 times 2 times 2 times 3 or using exponents for repeated multiplication of 2 it can be written as 2 to the fourth power times 3.\">\n<tbody>\n<tr>\n<td style=\"width: 70.14%;\">Ask yourself, &#8216;What prime number is a factor of [latex]48[\/latex]?&#8217;<\/p>\n<ul>\n<li>Well, the smallest prime number is [latex]2[\/latex].<\/li>\n<li>Divide [latex]48[\/latex] by [latex]2[\/latex], giving you [latex]24[\/latex].<\/li>\n<\/ul>\n<\/td>\n<td style=\"width: 29.0824%;\">\n<figure style=\"width: 293px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220024\/CNX_BMath_Figure_02_05_022_img-01.png\" alt=\".\" width=\"293\" height=\"108\" \/><figcaption class=\"wp-caption-text\">Factor Tree of 48(Partial)<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 70.14%;\">Next, consider [latex]24[\/latex].<\/p>\n<ul>\n<li>It&#8217;s not a prime number, so what two numbers multiply to get [latex]24[\/latex]?<\/li>\n<li>One option is [latex]4[\/latex] and [latex]6[\/latex].<\/li>\n<\/ul>\n<\/td>\n<td style=\"width: 29.0824%;\">\n<figure style=\"width: 293px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220025\/CNX_BMath_Figure_02_05_022_img-02.png\" alt=\".\" width=\"293\" height=\"181\" \/><figcaption class=\"wp-caption-text\">Factor Tree of 48 (Expanded)<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 70.14%;\">\n<ul>\n<li>Notice that [latex]4[\/latex] is not a prime number, but can be broken down further into [latex]2[\/latex] and [latex]2[\/latex].<\/li>\n<li>[latex]6[\/latex] can also be decomposed. Break it down into [latex]2[\/latex] and [latex]3[\/latex], with 3 being a prime number.<\/li>\n<\/ul>\n<\/td>\n<td style=\"width: 29.0824%;\">\n<figure style=\"width: 293px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24220026\/CNX_BMath_Figure_02_05_022_img-03.png\" alt=\".\" width=\"293\" height=\"266\" \/><figcaption class=\"wp-caption-text\">Factor Tree of 48 (Complete)<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 70.14%;\">We have fully factored down [latex]48[\/latex] into its prime factors.<br \/>\nLet&#8217;s write the product of all of the prime factors from least to greatest.<\/td>\n<td style=\"width: 29.0824%;\">[latex]48 = 2\\cdot 2\\cdot 2\\cdot 2\\cdot 3[\/latex]<\/p>\n<p>In <a class=\"glossary-term\" aria-haspopup=\"dialog\" aria-describedby=\"definition\" href=\"#term_401_756\">exponential form<\/a>: [latex]48 = 2^4 \\cdot 3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm18649\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=18649&theme=lumen&iframe_resize_id=ohm18649&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm18650\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=18650&theme=lumen&iframe_resize_id=ohm18650&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<div class=\"glossary\"><span class=\"screen-reader-text\" id=\"definition\">definition<\/span><template id=\"term_401_756\"><div class=\"glossary__definition\" role=\"dialog\" data-id=\"term_401_756\"><div tabindex=\"-1\"><p>Exponential notation is a way of expressing a number or function using a base raised to a power, which indicates how many times the base is multiplied by itself, such as [latex]a^n[\/latex] where [latex]a[\/latex] is the base and [latex]n[\/latex] is the exponent.<br \/>\nFor example: [latex]5\\cdot 5 = 5^2[\/latex].<\/p>\n<\/div><button><span aria-hidden=\"true\">&times;<\/span><span class=\"screen-reader-text\">Close definition<\/span><\/button><\/div><\/template><\/div>","protected":false},"author":12,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":32,"module-header":"background_you_need","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/401"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/users\/12"}],"version-history":[{"count":31,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/401\/revisions"}],"predecessor-version":[{"id":7951,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/401\/revisions\/7951"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/parts\/32"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapters\/401\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/media?parent=401"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=401"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/contributor?post=401"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/wp-json\/wp\/v2\/license?post=401"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}