What value of $k$ makes the statement true?
$\begin{array}{l}
x^k y^4\left(2 x^3+7 x^2 y^4\right)=2 x^4 y^4+7 x^3 y^8 \\
k=\square
\end{array}$
Answer (1)
Expand the left side of the equation using the distributive property: x k y 4 ( 2 x 3 + 7 x 2 y 4 ) = 2 x k + 3 y 4 + 7 x k + 2 y 8 .
Compare the expanded left side with the right side of the equation: 2 x k + 3 y 4 + 7 x k + 2 y 8 = 2 x 4 y 4 + 7 x 3 y 8 .
Equate the exponents of x in the corresponding terms: k + 3 = 4 and k + 2 = 3 .
Solve for k from either equation: k = 4 − 3 = 1 or k = 3 − 2 = 1 . Therefore, k = 1 .
Explanation
Understanding the Problem We are given the equation x k y 4 ( 2 x 3 + 7 x 2 y 4 ) = 2 x 4 y 4 + 7 x 3 y 8 We need to find the value of k that makes the equation true.
Expanding the Left Side First, let's expand the left side of the equation by distributing x k y 4 to both terms inside the parentheses: x k y 4 ( 2 x 3 + 7 x 2 y 4 ) = x k y 4 ⋅ 2 x 3 + x k y 4 ⋅ 7 x 2 y 4 = 2 x k + 3 y 4 + 7 x k + 2 y 4 + 4 = 2 x k + 3 y 4 + 7 x k + 2 y 8
Rewriting the Equation Now, we can rewrite the original equation with the expanded left side: 2 x k + 3 y 4 + 7 x k + 2 y 8 = 2 x 4 y 4 + 7 x 3 y 8
Comparing Exponents (First Term) To find the value of k , we can compare the exponents of the corresponding terms on both sides of the equation. Let's compare the exponents of x and y in the first terms: x k + 3 y 4 = x 4 y 4 This implies that k + 3 = 4 .
Comparing Exponents (Second Term) Now, let's compare the exponents of x and y in the second terms: x k + 2 y 8 = x 3 y 8 This implies that k + 2 = 3 .
Solving for k (First Equation) We can solve either equation to find the value of k . Let's solve the first equation: k + 3 = 4 Subtract 3 from both sides: k = 4 − 3 k = 1
Solving for k (Second Equation) Now, let's solve the second equation to verify our result: k + 2 = 3 Subtract 2 from both sides: k = 3 − 2 k = 1
Conclusion Both equations give us the same value for k , which is k = 1 . Therefore, the value of k that makes the statement true is 1.
Examples
Understanding exponents is crucial in many fields, such as computer science when dealing with data storage sizes (kilobytes, megabytes, gigabytes, etc.) or in physics when calculating quantities that scale exponentially, like radioactive decay. In finance, understanding exponential growth helps in calculating compound interest. For example, if you invest x dollars with an annual interest rate r compounded annually, after t years, the investment grows to x ( 1 + r ) t dollars. This formula relies on the principles of exponents and powers.
