To simplify the given expression [tex]\( 14 \left( 2 (x+y)^4 \right)^7 \)[/tex], let's break it down step-by-step.
1. Simplify the innermost expression:
[tex]\[
2 (x+y)^4
\][/tex]
2. Raise the result to the 7th power:
[tex]\[
\left( 2 (x+y)^4 \right)^7
\][/tex]
3. Apply the power to the constants and the expression separately:
Recall that when you have [tex]\( (ab)^n = a^n b^n \)[/tex]. Thus,
[tex]\[
\left( 2 (x+y)^4 \right)^7 = 2^7 \cdot \left( (x+y)^4 \right)^7
\][/tex]
4. Simplify each part:
- [tex]\( 2^7 \)[/tex] is [tex]\( 128 \)[/tex]
- [tex]\(( (x+y)^4 )^7 \)[/tex] is simplified using the power of a power property [tex]\( (a^m)^n = a^{m \cdot n} \)[/tex]:
[tex]\[
((x+y)^4)^7 = (x+y)^{4 \cdot 7} = (x+y)^{28}
\][/tex]
5. Combine these results:
[tex]\[
\left( 2 (x+y)^4 \right)^7 = 128 (x+y)^{28}
\][/tex]
6. Finally, multiply by 14:
[tex]\[
14 \cdot 128 (x+y)^{28}
\][/tex]
Calculate [tex]\( 14 \cdot 128 \)[/tex]:
[tex]\[
14 \times 128 = 1792
\][/tex]
7. Write the final expression:
[tex]\[
14 \left( 2 (x+y)^4 \right)^7 = 1792 (x+y)^{28}
\][/tex]
So, the simplified form of the expression is:
[tex]\[
1792 (x+y)^{28}
\][/tex]
