1. [latex]0.1\overline{5}[/latex] The bar above the [latex]5[/latex] indicates that it repeats. Therefore, [latex]0.1\overline{5}[/latex] is a repeating decimal, and is therefore a rational number.
2. [latex]4\pi[/latex] Since [latex]4\pi[/latex] is a multiple of pi, it is irrational. In this case, the rational part of the number is [latex]4[/latex], while the irrational part is [latex]\pi[/latex].
3. [latex]2.65987425\dots[/latex] The ellipsis [latex](\dots)[/latex] means that this number does not stop. There is no repeating pattern of digits. Since the number doesn’t stop and doesn’t repeat, it is irrational.

1. The square root of [latex]225[/latex] is [latex]15[/latex], which is an integer and can be expressed as a fraction [latex]\frac{15}{1}[/latex]. Therefore, [latex]\sqrt{225}[/latex] is rational.
2. The square root of [latex]5[/latex] is irrational, and multiplying it by [latex]3[/latex] doesn’t change that property. Therefore, [latex]3\sqrt{5}[/latex] is irrational.
3. The square root of [latex]80[/latex] can be simplified to [latex]4\sqrt{5}[/latex]. Since [latex]\sqrt{5}[/latex] is irrational, multiplying it by [latex]4[/latex] doesn’t change its irrationality. Therefore, [latex]\sqrt{80}[/latex] is also irrational.

Begin by finding the largest perfect square that is a factor of [latex]733[/latex]. We can do this by writing out the factor pairs of [latex]733[/latex]:

[latex]1 \times 733 \quad 11 \times 67[/latex]

Looking at the list of factors, there are no perfect squares other than [latex]1[/latex], which means [latex]\sqrt{733}[/latex] is already expressed in lowest terms. In this case, [latex]1[/latex] is the rational part, and [latex]\sqrt{733}[/latex] is the irrational part. Though we could write this as [latex]1\sqrt{733}[/latex], but the product of [latex]1[/latex] and any other number is just the number.

Begin by finding the largest perfect square that is a factor of [latex]1815[/latex]. We can do this by writing out the factor pairs of [latex]1815[/latex]:

[latex](1 \times 1815) \quad (3 \times 605) \quad (5 \times 363) \quad (7 \times 259) \quad (11 \times 165) \quad (13 \times 139) \quad (15 \times 121) \quad (21 \times 85) \quad (33 \times 55)[/latex]

Among these factors, the largest perfect square is [latex]121[/latex], so we factor the into [latex]121×15=11^2×15[/latex]. In the formula, [latex]a=11[/latex] and [latex]b=15[/latex]. Apply [latex]\sqrt{a^2 \times b}=\sqrt{a^2} \times \sqrt{b}[/latex].

[latex]\sqrt{11^2 \times 15}=\sqrt{11^2} \times \sqrt{15}[/latex]

The simplified form of [latex]\sqrt{1815}[/latex] is [latex]11\sqrt{15}[/latex]. In this example, the [latex]11[/latex] is the rational part, and the [latex]\sqrt{15}[/latex] is the irrational part.

Since these two irrational numbers have the same irrational part, [latex]\sqrt{15}[/latex], we can subtract without using a calculator. The rational part of the first number is [latex]41[/latex]. The rational part of the second number is [latex]23[/latex]. Using the formula yields:

[latex]41\sqrt{15}–23\sqrt{15}=(41-23) \times \sqrt{15}= 18\sqrt{15}[/latex]

Since these two irrational numbers have the same irrational part, [latex]π[/latex], the addition can be performed without using a calculator. The rational part of the first number is [latex]4.1[/latex]. The rational part of the second number is [latex]3.2[/latex]. Using the formula yields

[latex]41 \pi + 3.2 \pi = (4.1+3.2) \times \pi = 7.3\pi[/latex]

1. In this multiplication problem, [latex](1.2\sqrt{21})×(45\sqrt{14})[/latex], notice that the irrational parts of these numbers are similar. They are both square roots. Follow the process above.
   • Step 1: Multiply the rational parts. [latex]1.2×45=54[/latex]
   • Step 2: If necessary, reduce the result of Step 1 to lowest terms. This rational number is expressed as a decimal and will not be reduced.
   • Step 3: Multiply the irrational parts. [latex]\sqrt{21} \times \sqrt{14} = \sqrt{ 3 \times 12} = \sqrt{21 \times 14}[/latex]
   • Step 4: If necessary, reduce the result from Step 3 to lowest terms. [latex]\sqrt{21 \times 14} = \sqrt{294} = \sqrt{2 \times 147} = \sqrt{2 \times 3 \times 49} = \sqrt{2} \times \sqrt{3} \times \sqrt{49} = \sqrt{2} \times \sqrt{3} \times 7[/latex]
   • Step 5: The result is the product of the rational and irrational parts, which is [latex]54 \times \sqrt{2} \times \sqrt{3} \times 7 = 54 \times 7 \times \sqrt{2 \times 3} = 378 \sqrt{6}[/latex]
2. In this multiplication problem, [latex]38 \pi \times 2 \pi[/latex], notice that the irrational parts of these numbers are the same, [latex]π[/latex]. Follow the process above.
   • Step 1: Multiply the rational parts. [latex]38×2=76[/latex]
   • Step 2: If necessary, reduce the result of Step 1 to lowest terms. That result is an integer.
   • Step 3: Multiply the irrational parts. [latex]π×π=π^2[/latex]
   • Step 4: If necessary, reduce the result from Step 3 to lowest terms. This cannot be reduced.
   • Step 5: The result is the product of the rational and irrational parts, which is [latex]76π^2[/latex].

In this division problem, [latex](84\sqrt{132}) \div (14\sqrt{11})[/latex], notice that the irrational parts of these numbers are similar. They are both square roots, so follow the steps given above.

• Step 1: Divide the rational parts. [latex]\frac{84}{14} = 6[/latex]
• Step 2: If necessary, reduce the result of Step 1 to lowest terms. The number 6 is already in lowest terms.
• Step 3: Divide the irrational parts. [latex]\frac{\sqrt{132}}{\sqrt{11}} = \sqrt{\frac{132}{11}}[/latex]
• Step 4: If necessary, reduce the result from Step 3 to lowest terms. [latex]\sqrt{\frac{132}{11}} = \sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}[/latex]
• Step 5: The result is the product of the rational and irrational parts, which is [latex]6 \times 2\sqrt{3} = 12\sqrt{3}[/latex].