Consider the following algebraic equation, where x \neq 0: \frac{(3x^4)^2}{9x^k} = x^5 Wha… — Armed Services Vocational Aptitude Battery (ASVAB)

Consider the following algebraic equation, where $x \neq 0$ :

\frac{(3x^4)^2}{9x^k} = x^5

What is the value of the exponent $k$ ?

Answer: 3 / k=3 / k = 3 / three

First, the numerator

(3x^4)^2

simplifies to

9x^8

. Substituting this back yields

\frac{9x^8}{9x^k} = x^5

. The coefficients cancel out, and applying the quotient rule to the variables gives

x^{8-k} = x^5

. By equating the exponents, we get

8 - k = 5

, which means

k = 3

Apply the power of a product rule to the numerator.

(3x^4)^2 = 3^2 \cdot (x^4)^2 = 9x^8

When raising a product to a power, you must raise every factor inside the parentheses to that power.

Substitute the simplified numerator back into the equation and simplify the numerical coefficients.

\frac{9x^8}{9x^k} = \frac{x^8}{x^k}

The coefficients

\frac{9}{9}

cancel out to 1, making it easier to isolate the variable base.

Apply the quotient rule for exponents on the left side of the equation.

\frac{x^8}{x^k} = x^{8-k}

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

Set the resulting exponent equal to the exponent on the right side of the original equation and solve for

k

8 - k = 5 \Rightarrow k = 3

Since the bases on both sides of the equation are the same (

x

), their exponents must be equal.

Operations with Variable Exponents

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