Problem Solution:
To simplify the expression [tex]\(\frac{(2x + 10)^4}{(x + 5)^3}\)[/tex], we will break it down step by step.
### Step-by-Step Solution:
1. Factor the numerator and the denominator:
[tex]\[
(2x + 10)^4 = [(2x + 10)(2x + 10)(2x + 10)(2x + 10)]
\][/tex]
[tex]\[
(x + 5)^3 = [(x + 5)(x + 5)(x + 5)]
\][/tex]
Notice that [tex]\(2x + 10\)[/tex] can be rewritten as [tex]\(2(x + 5)\)[/tex]. So, let’s rewrite the numerator:
[tex]\[
(2(x + 5))^4
\][/tex]
2. Simplify the exponent:
By applying the power rule [tex]\((ab)^n = a^n b^n\)[/tex]:
[tex]\[
(2(x + 5))^4 = 2^4 \cdot (x + 5)^4
\][/tex]
3. Simplify the powers:
Calculate [tex]\(2^4\)[/tex]:
[tex]\[
2^4 = 16
\][/tex]
So, the numerator simplifies to:
[tex]\[
16(x + 5)^4
\][/tex]
4. Combine the simplified numerator with the denominator:
[tex]\[
\frac{16(x + 5)^4}{(x + 5)^3}
\][/tex]
5. Simplify the fraction:
We know that [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]. Applying this:
[tex]\[
\frac{(x + 5)^4}{(x + 5)^3} = (x + 5)^{4-3} = (x + 5)^1 = x + 5
\][/tex]
6. Put it all together:
[tex]\[
16 \cdot (x + 5)
\][/tex]
Therefore, the simplified expression is:
[tex]\[
16(x + 5)
\][/tex]
Let’s expand this expression to show the final answer:
[tex]\[
16(x + 5) = 16x + 80
\][/tex]
So the simplified form of the given expression [tex]\(\frac{(2x + 10)^4}{(x + 5)^3}\)[/tex] is:
[tex]\[
\boxed{16x + 80}
\][/tex]
