For all non-zero real numbers x, the expression x^4 \cdot (x^3)^k is equivalent to x^{10}.… — ACT

For all non-zero real numbers $x$ , the expression $x^4 \cdot (x^3)^k$ is equivalent to $x^{10}$ . What is the value of the integer $k$ ?

Answer: 2

The correct answer is 2.

First, use the power of a power property to write

(x^3)^k

x^{3k}

. The expression then becomes

x^4 \cdot x^{3k}

. Next, use the product of powers property to combine the terms into

x^{4+3k}

. Since the expression is equivalent to

x^{10}

, set the exponents equal:

4 + 3k = 10

. Solving this equation gives

3k = 6

, which simplifies to

k = 2

Apply the power of a power rule

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

to simplify

(x^3)^k

x^{3k}

To raise a power to another power, multiply the exponents.

Apply the product of powers rule

x^a \cdot x^b = x^{a+b}

to combine the terms

x^4 \cdot x^{3k}

x^{4+3k}

When multiplying exponential terms with the same base, add their exponents.

Set the combined exponent

4+3k

equal to the target exponent

10

and solve for

k

k = 2

Since the bases are equal and non-zero, their exponents must be equal.

Properties of exponents in algebraic expressions (power of a power rule and product of powers rule)

Estimated Time:45s

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