Question

1. which is equivalent to the expression below?

$a^{\frac{5}{4}}b^{\frac{13}{4}}$
a. $\frac{1}{4}\sqrt{a^5b^{13}}$
c. $a^2b^6\sqrt4{ab}$
b. $a^4b^4\sqrt{ab}$
d. $ab^3\sqrt4{ab}$

Explanation:

Step1: Rewrite exponents as integers + remainders

For $a$: $\frac{5}{4} = 1 + \frac{1}{4}$; For $b$: $\frac{13}{4} = 3 + \frac{1}{4}$
Expression: $a^{1+\frac{1}{4}}b^{3+\frac{1}{4}} = a^1a^{\frac{1}{4}}b^3b^{\frac{1}{4}}$

Step2: Group integer and fractional exponents

Combine terms with integer exponents and fractional exponents separately.
Expression: $ab^3 \cdot a^{\frac{1}{4}}b^{\frac{1}{4}} = ab^3 \cdot (ab)^{\frac{1}{4}}$

Step3: Convert fractional exponent to radical

Recall $x^{\frac{1}{n}} = \sqrt[n]{x}$, so $(ab)^{\frac{1}{4}} = \sqrt[4]{ab}$
Expression: $ab^3\sqrt[4]{ab}$

Answer:

D. $ab^3\sqrt[4]{ab}$