{"id":576,"date":"2024-04-23T01:10:40","date_gmt":"2024-04-23T01:10:40","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=576"},"modified":"2024-11-20T00:52:43","modified_gmt":"2024-11-20T00:52:43","slug":"exponents-and-scientific-notation-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/exponents-and-scientific-notation-learn-it-3\/","title":{"raw":"Exponents and Scientific Notation: Learn It 3","rendered":"Exponents and Scientific Notation: Learn It 3"},"content":{"raw":"

Raise Powers to Powers<\/h2>\r\n

The Power Rule for Exponents<\/h3>\r\nAnother word for an exponent is power. You have likely seen or heard an example such as [latex]3^5[\/latex] can be described as [latex]3[\/latex] raised to the [latex]5[\/latex]th power. In this section, we will further expand our capabilities with exponents. We will learn what to do when a term with a power is raised to another power, what to do when two numbers or variables are multiplied and both are raised to an exponent, and what to do when numbers or variables that are divided are raised to a power. We will begin by raising powers to powers.\r\n\r\n
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the power rule for exponents<\/h3>\r\nFor any positive number [latex]x[\/latex] and integers [latex]a[\/latex] and [latex]b[\/latex]: [latex]\\left(x^{a}\\right)^{b}=x^{a\\cdot{b}}[\/latex].\r\n\r\n<\/div>\r\n<\/section>
[ohm2_question hide_question_numbers=1]6821[\/ohm2_question]<\/section>\r\n

Raise a Product to a Power<\/h3>\r\nRaising a product to a power is a fundamental operation in algebra that demonstrates how exponents interact with multiplication. This operation is widely used across various mathematical disciplines, including geometry, where it might be used to calculate the volume of shapes, and in finance, where it can be used to calculate compounded interest over multiple periods.\r\n\r\nThe rule simplifies the process of working with powers of products. Instead of multiplying the base numbers repeatedly, we apply the exponent to each factor individually. This is based on the distributive property of exponents over multiplication.\r\n\r\n
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a product raised to a power<\/h3>\r\nFor any nonzero numbers [latex]a[\/latex] and [latex]b[\/latex] and any integer [latex]x[\/latex], [latex]\\left(ab\\right)^{x}=a^{x}\\cdot{b^{x}}[\/latex].\r\n\r\n<\/div>\r\n<\/section>
Simplify the following:\u00a0
[latex]\\left(2yz\\right)^{6}[\/latex]<\/center>[reveal-answer q=\"368657\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"368657\"]Apply the exponent to each number in the product.[latex]2^{6}y^{6}z^{6}[\/latex]\r\n
Answer: [latex]\\left(2yz\\right)^{6}=64y^{6}z^{6}[\/latex]<\/center>[\/hidden-answer]<\/section>If the variable has an exponent with it, use the Power Rule: multiply the exponents.\r\n\r\n
Simplify the following:
[latex]\\left(\u22127a^{4}b\\right)^{2}[\/latex]<\/center>[reveal-answer q=\"136794\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"136794\"]Apply the exponent [latex]2[\/latex]\u00a0to each factor within the parentheses.[latex]\\left(\u22127\\right)^{2}\\left(a^{4}\\right)^{2}\\left(b\\right)^{2}[\/latex]Square the coefficient and use the Power Rule to square\u00a0[latex]\\left(a^{4}\\right)^{2}[\/latex].\r\n

[latex]49a^{4\\cdot2}b^{2}[\/latex]<\/p>\r\nSimplify.\r\n

[latex]49a^{8}b^{2}[\/latex]<\/p>\r\n \r\n\r\n

Answer: [latex]\\left(-7a^{4}b\\right)^{2}=49a^{8}b^{2}[\/latex]<\/center>[\/hidden-answer]\r\n\r\n<\/section>\r\n

Raise a Quotient to a Power<\/h3>\r\nRaising a quotient to a power is another key concept in algebra that involves exponents. This operation is essential when dealing with division in the context of exponential expressions. The power of a quotient rule tells us that when you raise a quotient to an exponent, you raise both the numerator and the denominator to that exponent separately.\r\n\r\nThis rule is incredibly useful for simplifying complex algebraic expressions, solving equations, and understanding geometric growth or decay when dealing with fractions or ratios\r\n\r\n
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a quotient raised to a power<\/h3>\r\nFor any number [latex]a[\/latex], any non-zero number [latex]b[\/latex], and any integer [latex]x[\/latex], [latex] \\displaystyle {\\left(\\frac{a}{b}\\right)}^{x}=\\frac{a^{x}}{b^{x}}[\/latex].\r\n\r\n<\/div>\r\n<\/section>
Simplify the following:
[latex] \\displaystyle {{\\left( \\frac{2{x}^{2}y}{x} \\right)}^{3}}[\/latex]<\/center>[reveal-answer q=\"875425\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"875425\"]Apply the power to each factor individually.\r\n

[latex] \\displaystyle \\frac{{{2}^{3}{\\left({x}^{2}\\right)}^{3}{y}^{3}}}{{{x}^{3}}}[\/latex]<\/p>\r\nSeparate into numerical and variable factors.\r\n

[latex] \\displaystyle {{2}^{3}}\\cdot \\frac{{{x}^{3\\cdot2}}}{{{x}^{3}}}\\cdot \\frac{{{y}^{3}}}{1}[\/latex]<\/p>\r\nSimplify by taking [latex]2[\/latex] to the third power and applying the Power and Quotient Rules for exponents\u2014multiply\u00a0and subtract the exponents of matching variables.\r\n

[latex] \\displaystyle 8\\cdot {{x}^{(6-3)}}\\cdot {{y}^{3}}[\/latex]<\/p>\r\nSimplify.\r\n

[latex] \\displaystyle 8{{x}^{3}}{{y}^{3}}[\/latex]<\/p>\r\n \r\n\r\n

Answer: [latex] \\displaystyle {{\\left( \\frac{2{x}^{2}y}{x} \\right)}^{3}}=8{{x}^{3}}{{y}^{3}}[\/latex]<\/center>[\/hidden-answer]\r\n\r\n<\/section>","rendered":"

Raise Powers to Powers<\/h2>\n

The Power Rule for Exponents<\/h3>\n

Another word for an exponent is power. You have likely seen or heard an example such as [latex]3^5[\/latex] can be described as [latex]3[\/latex] raised to the [latex]5[\/latex]th power. In this section, we will further expand our capabilities with exponents. We will learn what to do when a term with a power is raised to another power, what to do when two numbers or variables are multiplied and both are raised to an exponent, and what to do when numbers or variables that are divided are raised to a power. We will begin by raising powers to powers.<\/p>\n

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the power rule for exponents<\/h3>\n

For any positive number [latex]x[\/latex] and integers [latex]a[\/latex] and [latex]b[\/latex]: [latex]\\left(x^{a}\\right)^{b}=x^{a\\cdot{b}}[\/latex].<\/p>\n<\/div>\n<\/section>\n