If x and y are positive real numbers such that (x^3 y^{-2})^k \cdot (x^{-1} y^4)^3 = x^6 y… — ACT

x

and

y

are positive real numbers such that

(x^3 y^{-2})^k \cdot (x^{-1} y^4)^3 = x^6 y^6

what is the value of the exponent

k

Answer: 3

The value of the exponent

k

is 3.

Applying the exponent rules simplifies the left side of the equation to

x^{3k-3}y^{12-2k}

. Equating the exponent of

x

to the right side gives

3k - 3 = 6

, which yields

k = 3

. This value is confirmed by equating the exponent of

y

, since

12 - 2(3) = 6

Apply the power of a power rule

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

to expand the terms in the expression.

(x^3 y^{-2})^k = x^{3k} y^{-2k}

and

(x^{-1} y^4)^3 = x^{-3} y^{12}

To remove the outer parentheses by multiplying the internal exponents of each variable by the outer exponent.

Multiply the terms together by applying the product rule for exponents,

a^m \cdot a^n = a^{m+n}

(x^{3k} y^{-2k})(x^{-3} y^{12}) = x^{3k-3} y^{12-2k}

To combine the like bases of

x

and

y

into a single simplified expression.

Set the exponents of like bases equal to the exponents on the right-hand side of the equation,

x^6 y^6

3k - 3 = 6

and

12 - 2k = 6

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

Solve the linear equation

3k - 3 = 6

for

k

3k = 9 \implies k = 3

To determine the numerical value of the variable

k

Verify the solution by solving the second linear equation,

12 - 2k = 6

-2k = -6 \implies k = 3

To ensure consistency across both variable exponents in the expression.

Properties of Exponents in Algebraic Expressions

Question