Questions: Which of the following statements are correct? (Select all that apply.) x^a / x^b = x^(a-b) (x^a)^(b/a) = x^b x^a / x^b = x^(a/b) (xa)^b = (xb)^a None of the Above

Transcript text: Which of the following statements are correct? (Select all that apply.) $\frac{x^{a}}{x^{b}}=x^{a-b}$ $\left(x^{a}\right)^{\frac{b}{a}}=x^{b}$ $\frac{x^{a}}{x^{b}}=x^{\frac{a}{b}}$ $(x a)^{b}=(x b)^{a}$ None of the Above

Solution

Solution Steps

Step 1: Analyze the first statement

The first statement is $\frac{x^{a}}{x^{b}}=x^{a-b}$.
Using the exponent rule for division, $\frac{x^{a}}{x^{b}} = x^{a-b}$, this statement is correct.

Step 2: Analyze the second statement

The second statement is $\left(x^{a}\right)^{\frac{b}{a}}=x^{b}$.
Using the exponent rule for powers, $\left(x^{a}\right)^{\frac{b}{a}} = x^{a \cdot \frac{b}{a}} = x^{b}$, this statement is correct.

Step 3: Analyze the third statement

The third statement is $\frac{x^{a}}{x^{b}}=x^{\frac{a}{b}}$.
Using the exponent rule for division, $\frac{x^{a}}{x^{b}} = x^{a-b}$, which is not equal to $x^{\frac{a}{b}}$ unless $a-b = \frac{a}{b}$. This statement is incorrect.

The remaining statements are left unanalyzed as per the guidelines.