The algebraic expression \frac{(x^2)^a}{x^{-4}} is equivalent to x^{10} for all non-zero r… — ACT

The algebraic expression $\frac{(x^2)^a}{x^{-4}}$ is equivalent to $x^{10}$ for all non-zero real numbers $x$ . What is the value of $a$ ?

Answer: 3

The correct answer is 3.

The correct value for

a

3

. Applying the power of a power rule to

(x^2)^a

yields

x^{2a}

. Then, applying the quotient rule to

\frac{x^{2a}}{x^{-4}}

yields

x^{2a - (-4)} = x^{2a+4}

. Equating the exponents gives

2a + 4 = 10

, which solves to

a = 3

Apply the power of a power property to the numerator.

(x^2)^a = x^{2a}

When raising a power to another power, multiply the exponents:

(x^m)^n = x^{mn}

Apply the quotient property of exponents to simplify the fraction.

\frac{x^{2a}}{x^{-4}} = x^{2a - (-4)} = x^{2a + 4}

When dividing exponential expressions with the same base, subtract the exponent in the denominator from the exponent in the numerator:

\frac{x^m}{x^n} = x^{m-n}

Set the resulting exponent equal to the exponent of the equivalent expression and solve for

a

2a + 4 = 10 \implies 2a = 6 \implies a = 3

Since the bases are identical and the expressions are equivalent, their exponents must be equal.

Properties of Exponents in Algebraic Expressions

Question