Sure! Let's analyze and simplify the given expression step-by-step:
Given expression:
[tex]\[
\left(x^{10} + y^5\right)^{15}
\][/tex]
This is a power expression where the base [tex]\((x^{10} + y^5)\)[/tex] is raised to the 15th power. To simplify or expand this expression, you need to understand that it involves terms raised to a high power.
1. Identify the Base and the Exponent:
- The base in this expression is [tex]\(x^{10} + y^5\)[/tex].
- The exponent is 15.
2. Understand the Expansion:
- If you were to expand this using the binomial theorem, it would look very complex because the binomial theorem states:
[tex]\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\][/tex]
- In your case, [tex]\(a = x^{10}\)[/tex], [tex]\(b = y^5\)[/tex], and [tex]\(n = 15\)[/tex]. The expansion would involve terms like:
[tex]\[
\sum_{k=0}^{15} \binom{15}{k} (x^{10})^{15-k} (y^5)^k
\][/tex]
- Each term in the expansion would be of the form:
[tex]\[
\binom{15}{k} x^{10(15-k)} y^{5k}
\][/tex]
However, without going into the full expansion since it is extremely lengthy, you can express the result as:
[tex]\[
(x^{10} + y^5)^{15}
\][/tex]
This is a more compact form and is a correct representation of the expression raised to the 15th power.
