For x \neq 0, the expression \frac{(x^2)^5}{x^k} simplifies to x^6. What is the value of t… — ACT

For $x \neq 0$ , the expression $\frac{(x^2)^5}{x^k}$ simplifies to $x^6$ . What is the value of the exponent $k$ ?

Answer: 4

The value of the exponent

k

is 4.

Applying the power of a power rule to the numerator yields

(x^2)^5 = x^{10}

. Next, applying the quotient rule to divide

x^{10}

x^k

yields

x^{10-k}

. Setting this equal to the simplified term

x^6

leads to the exponent equation

10 - k = 6

. Solving this equation gives the final result

k = 4

Simplify the numerator expression

(x^2)^5

x^{10}

Multiply the exponents when raising a power to another power:

(x^a)^b = x^{ab}

Simplify the division of the two exponential expressions

\frac{x^{10}}{x^k}

x^{10-k}

Subtract the exponent of the denominator from the exponent of the numerator:

\frac{x^a}{x^b} = x^{a-b}

Solve the linear equation for

k

using the target exponent 6

k = 4

Equating the exponent

10-k

6

gives

10-k=6

. Subtracting 10 from both sides gives

-k = -4

, so

k = 4

Applying properties of exponents in algebraic expressions, specifically the power of a power rule and the quotient rule.

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