Question

1. ( left(64x^{3}y^{5}

ight)^{\frac{1}{4}} )

Explanation:

Step1: Apply exponent product rule

$(ab)^n=a^n b^n$
So, $(64x^3y^5)^{\frac{1}{4}} = 64^{\frac{1}{4}} \cdot (x^3)^{\frac{1}{4}} \cdot (y^5)^{\frac{1}{4}}$

Step2: Simplify constant term

Rewrite 64 as $2^6$, so:
$64^{\frac{1}{4}}=(2^6)^{\frac{1}{4}}=2^{\frac{6}{4}}=2^{\frac{3}{2}}$

Step3: Simplify variable terms

Use $(a^m)^n=a^{mn}$:
$(x^3)^{\frac{1}{4}}=x^{\frac{3}{4}}$, $(y^5)^{\frac{1}{4}}=y^{\frac{5}{4}}$

Step4: Combine all terms

$2^{\frac{3}{2}}x^{\frac{3}{4}}y^{\frac{5}{4}}$ can also be written as $\sqrt{8}x^{\frac{3}{4}}y^{\frac{5}{4}}$ or $2\sqrt{2}x^{\frac{3}{4}}y^{\frac{5}{4}}$

Answer:

$2^{\frac{3}{2}}x^{\frac{3}{4}}y^{\frac{5}{4}}$ (or equivalent form $2\sqrt{2}x^{\frac{3}{4}}y^{\frac{5}{4}}$)