Answer :
To determine the value of [tex]\( k \)[/tex] that makes the statement true, we want to simplify the expression on the left-hand side and then match it to the terms on the right-hand side.
Consider the expression:
[tex]\[ x^k y^4 \left(2x^3 + 7x^2 y^4\right) \][/tex]
Let's distribute [tex]\( x^k y^4 \)[/tex] across the terms inside the parentheses:
1. Multiply [tex]\( x^k y^4 \)[/tex] by the first term [tex]\( 2x^3 \)[/tex]:
[tex]\[
x^k y^4 \cdot 2x^3 = 2x^{k+3} y^4
\][/tex]
This expression simplifies as the powers of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] add up.
2. Multiply [tex]\( x^k y^4 \)[/tex] by the second term [tex]\( 7x^2 y^4 \)[/tex]:
[tex]\[
x^k y^4 \cdot 7x^2 y^4 = 7x^{k+2} y^8
\][/tex]
Here too, we add the exponents for both [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
Now, the expanded expression is:
[tex]\[ 2x^{k+3} y^4 + 7x^{k+2} y^8 \][/tex]
We need this expression to equal the given expression on the right-hand side:
[tex]\[ 2x^4 y^4 + 7x^3 y^8 \][/tex]
Let's equate the terms:
- For the first term [tex]\( 2x^{k+3} y^4 = 2x^4 y^4 \)[/tex]:
- The exponents of [tex]\( x \)[/tex] should match:
[tex]\[
k + 3 = 4
\][/tex]
- Solving for [tex]\( k \)[/tex]:
[tex]\[
k = 1
\][/tex]
- For the second term [tex]\( 7x^{k+2} y^8 = 7x^3 y^8 \)[/tex]:
- The exponents of [tex]\( x \)[/tex] should match:
[tex]\[
k + 2 = 3
\][/tex]
- Solving for [tex]\( k \)[/tex]:
[tex]\[
k = 1
\][/tex]
Both terms indicate that [tex]\( k = 1 \)[/tex]. Therefore, the value of [tex]\( k \)[/tex] that makes the statement true is [tex]\( k = 1 \)[/tex].
Consider the expression:
[tex]\[ x^k y^4 \left(2x^3 + 7x^2 y^4\right) \][/tex]
Let's distribute [tex]\( x^k y^4 \)[/tex] across the terms inside the parentheses:
1. Multiply [tex]\( x^k y^4 \)[/tex] by the first term [tex]\( 2x^3 \)[/tex]:
[tex]\[
x^k y^4 \cdot 2x^3 = 2x^{k+3} y^4
\][/tex]
This expression simplifies as the powers of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] add up.
2. Multiply [tex]\( x^k y^4 \)[/tex] by the second term [tex]\( 7x^2 y^4 \)[/tex]:
[tex]\[
x^k y^4 \cdot 7x^2 y^4 = 7x^{k+2} y^8
\][/tex]
Here too, we add the exponents for both [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
Now, the expanded expression is:
[tex]\[ 2x^{k+3} y^4 + 7x^{k+2} y^8 \][/tex]
We need this expression to equal the given expression on the right-hand side:
[tex]\[ 2x^4 y^4 + 7x^3 y^8 \][/tex]
Let's equate the terms:
- For the first term [tex]\( 2x^{k+3} y^4 = 2x^4 y^4 \)[/tex]:
- The exponents of [tex]\( x \)[/tex] should match:
[tex]\[
k + 3 = 4
\][/tex]
- Solving for [tex]\( k \)[/tex]:
[tex]\[
k = 1
\][/tex]
- For the second term [tex]\( 7x^{k+2} y^8 = 7x^3 y^8 \)[/tex]:
- The exponents of [tex]\( x \)[/tex] should match:
[tex]\[
k + 2 = 3
\][/tex]
- Solving for [tex]\( k \)[/tex]:
[tex]\[
k = 1
\][/tex]
Both terms indicate that [tex]\( k = 1 \)[/tex]. Therefore, the value of [tex]\( k \)[/tex] that makes the statement true is [tex]\( k = 1 \)[/tex].
