For all non-zero real numbers a and b, the expression (a^2 b^k)^3 is equivalent to a^6 b^{… — ACT

For all non-zero real numbers $a$ and $b$ , the expression $(a^2 b^k)^3$ is equivalent to $a^6 b^{15}$ . What is the value of the integer $k$ ?

Answer: 5

The value of the integer

k

is 5.

To find the value of

k

, we simplify the expression

(a^2 b^k)^3

using exponent rules. According to the power of a product property,

(xy)^z = x^z y^z

, so

(a^2 b^k)^3 = (a^2)^3 (b^k)^3

. Next, applying the power of a power property,

(x^y)^z = x^{yz}

, we get

a^{2 \cdot 3} b^{k \cdot 3} = a^6 b^{3k}

. Since this expression is equivalent to

a^6 b^{15}

, we set the exponents of

b

equal to each other:

3k = 15

. Dividing both sides by 3 yields

k = 5

Apply the power of a product rule to the expression

(a^2 b^k)^3

(a^2)^3 \cdot (b^k)^3

The power of a product rule states that

(xy)^z = x^z y^z

Apply the power of a power rule to simplify the exponents.

a^6 b^{3k}

The power of a power rule states that

(x^y)^z = x^{y \cdot z}

, so

(a^2)^3 = a^{2 \cdot 3} = a^6

and

(b^k)^3 = b^{3k}

Set the exponent of

b

a^6 b^{3k}

equal to the exponent of

b

in the equivalent expression

a^6 b^{15}

3k = 15

Since the expressions are equivalent for all non-zero real numbers, the exponents of like bases must be equal.

Solve the linear equation for

k

k = 5

Dividing both sides of

3k = 15

by 3 isolates the variable

k

Properties of exponents, specifically the power of a product rule

(xy)^z = x^z y^z

and the power of a power rule

(x^y)^z = x^{y \cdot z}

Estimated Time:45s

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