Question

what value of \\(k\\) makes the statement true?
\\x^k y^4 (2x^3 + 7x^2 y^4) = 2x^4 y^4 + 7x^3 y^8\\
\\(k =\\)

Explanation:

Apply the distributive property

We distribute the monomial \(x^k y^4\) to each term inside the parentheses.
Using the Distributive Property knowledge point:

$$ x^k y^4 (2x^3 + 7x^2 y^4) = (x^k y^4)(2x^3) + (x^k y^4)(7x^2 y^4) $$

Simplify each term

We multiply the coefficients and add the exponents of like bases.
Using the Product of Powers knowledge point:

$$ (x^k y^4)(2x^3) = 2x^{k+3} y^4 $$

$$ (x^k y^4)(7x^2 y^4) = 7x^{k+2} y^{4+4} = 7x^{k+2} y^8 $$

Thus, the expanded expression is:

$$ 2x^{k+3} y^4 + 7x^{k+2} y^8 $$

Equate to the given right-hand side

We set our expanded expression equal to the given right-hand side.

$$ 2x^{k+3} y^4 + 7x^{k+2} y^8 = 2x^4 y^4 + 7x^3 y^8 $$

Solve for k

By comparing the corresponding exponents of \(x\) on both sides, we set up an equation.

$$ k + 3 = 4 \implies k = 1 $$

We verify with the second term:

$$ k + 2 = 3 \implies k = 1 $$

Both terms yield the same value.

Answer:

What value of \(k\) makes the statement true?
\(x^k y^4(2x^3 + 7x^2 y^4) = 2x^4 y^4 + 7x^3 y^8\)
\(k =\) <blank>1</blank>