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)
The problem requires finding the value of k that satisfies the equation x k y 4 ( 2 x 3 + 7 x 2 y 4 ) = 2 x 4 y 4 + 7 x 3 y 8 . The approach involves expanding the left side, comparing the exponents of x on both sides, and solving for k . The steps are as follows:
Expand the left side of the equation: 2 x k + 3 y 4 + 7 x k + 2 y 8 .
Equate the exponents of x in the first terms: k + 3 = 4 .
Solve for k : k = 1 .
Verify by equating the exponents of x in the second terms: k + 2 = 3 , which also gives k = 1 .
The final answer is 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 and 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 (Term 1) To find the value of k , we can compare the exponents of x and y on both sides of the equation. Let's compare the exponents of x in the first terms: k + 3 = 4
Solving for k (Term 1) Now, let's solve for k :
k = 4 − 3 k = 1
Comparing Exponents (Term 2) We can also compare the exponents of x in the second terms to verify our answer: k + 2 = 3
Solving for k (Term 2) Solving for k :
k = 3 − 2 k = 1
Conclusion Since we obtained the same value for k from both terms, we can conclude that the value of k that makes the statement true is 1.
k = 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. For instance, if the size of data doubles every year, the exponent helps determine the size after a certain number of years. Similarly, in finance, compound interest calculations rely heavily on understanding exponents to determine the growth of investments over time.
