Question

1. simplify each radical expression by extracting roots if possible (+6 pts).

a) $25^{\frac{6}{5}}$
b) $sqrt{108}$
c) $108^{1/3}$

Explanation:

Step1: Rewrite as radical form

$25^{\frac{6}{5}} = \sqrt[5]{25^6}$

Step2: Split exponent into multiple of 5

$\sqrt[5]{25^5 \times 25} = \sqrt[5]{25^5} \times \sqrt[5]{25}$

Step3: Simplify the perfect 5th root

$25 \times \sqrt[5]{25} = 25 \times \sqrt[5]{5^2}$
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Step1: Factor 108 into perfect square

$\sqrt{108} = \sqrt{36 \times 3}$

Step2: Split into product of radicals

$\sqrt{36} \times \sqrt{3}$

Step3: Simplify perfect square root

$6 \times \sqrt{3}$
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Step1: Rewrite as cube root form

$108^{1/3} = \sqrt[3]{108}$

Step2: Factor 108 into perfect cube

$\sqrt[3]{27 \times 4}$

Step3: Split into product of radicals

$\sqrt[3]{27} \times \sqrt[3]{4}$

Step4: Simplify perfect cube root

$3 \times \sqrt[3]{4}$

Answer:

a) $25\sqrt[5]{25}$ (or $25 \times 5^{\frac{2}{5}}$)
b) $6\sqrt{3}$
c) $3\sqrt[3]{4}$