Question

1. $(16x^{8})^{1/4}$
2. $(64x^{12})^{1/6}$
3. $(125x^{15})^{-1/5}$
4. $(81x^{8})^{1/4}$
5. $(27x^{9})^{1/3}$
6. $(256x^{16})^{1/4}$

Explanation:

Problem 3: $(16x^8)^{1/4}$

Step1: Apply exponent product rule

$(ab)^n=a^n b^n$
$(16x^8)^{1/4}=16^{1/4} \cdot (x^8)^{1/4}$

Step2: Simplify constant and variable

$16^{1/4}=2$, $(x^8)^{1/4}=x^{8 \cdot \frac{1}{4}}=x^2$
$2 \cdot x^2$

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Problem 4: $(64x^{12})^{1/6}$

Step1: Apply exponent product rule

$(ab)^n=a^n b^n$
$(64x^{12})^{1/6}=64^{1/6} \cdot (x^{12})^{1/6}$

Step2: Simplify constant and variable

$64^{1/6}=2$, $(x^{12})^{1/6}=x^{12 \cdot \frac{1}{6}}=x^2$
$2 \cdot x^2$

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Problem 5: $(125x^{15})^{-1/5}$

Step1: Apply negative exponent rule

$a^{-n}=\frac{1}{a^n}$
$(125x^{15})^{-1/5}=\frac{1}{(125x^{15})^{1/5}}$

Step2: Apply exponent product rule

$(ab)^n=a^n b^n$
$\frac{1}{125^{1/5} \cdot (x^{15})^{1/5}}$

Step3: Simplify constant and variable

$125^{1/5}=5^{3/5}$, $(x^{15})^{1/5}=x^3$
$\frac{1}{5^{3/5}x^3}$

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Problem 6: $(81x^8)^{-1/4}$

Step1: Apply negative exponent rule

$a^{-n}=\frac{1}{a^n}$
$(81x^8)^{-1/4}=\frac{1}{(81x^8)^{1/4}}$

Step2: Apply exponent product rule

$(ab)^n=a^n b^n$
$\frac{1}{81^{1/4} \cdot (x^8)^{1/4}}$

Step3: Simplify constant and variable

$81^{1/4}=3$, $(x^8)^{1/4}=x^2$
$\frac{1}{3x^2}$

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Problem 7: $(27x^9)^{1/3}$

Step1: Apply exponent product rule

$(ab)^n=a^n b^n$
$(27x^9)^{1/3}=27^{1/3} \cdot (x^9)^{1/3}$

Step2: Simplify constant and variable

$27^{1/3}=3$, $(x^9)^{1/3}=x^{9 \cdot \frac{1}{3}}=x^3$
$3 \cdot x^3$

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Problem 8: $(256x^{16})^{1/4}$

Step1: Apply exponent product rule

$(ab)^n=a^n b^n$
$(256x^{16})^{1/4}=256^{1/4} \cdot (x^{16})^{1/4}$

Step2: Simplify constant and variable

$256^{1/4}=4$, $(x^{16})^{1/4}=x^{16 \cdot \frac{1}{4}}=x^4$
$4 \cdot x^4$

Answer:

1. $2x^2$
2. $2x^2$
3. $\frac{1}{5^{3/5}x^3}$
4. $\frac{1}{3x^2}$
5. $3x^3$
6. $4x^4$