The given expression is \((3a^5x + 5y^{x+1})^2\).

This is a binomial raised to the power of 2, so we can apply the **square of a binomial** formula:

\[
(a + b)^2 = a^2 + 2ab + b^2
\]

Here, \(a = 3a^5x\) and \(b = 5y^{x+1}\).

Now, let's expand the expression step by step:

1. Square the first term \( (3a^5x)^2 \):
   \[
   (3a^5x)^2 = 9a^{10}x^2
   \]

2. Multiply the two terms and then double it \(2(3a^5x)(5y^{x+1})\):
   \[
   2(3a^5x)(5y^{x+1}) = 30a^5x y^{x+1}
   \]

3. Square the second term \( (5y^{x+1})^2 \):
   \[
   (5y^{x+1})^2 = 25y^{2(x+1)} = 25y^{2x+2}
   \]

Now, putting it all together:

\[
(3a^5x + 5y^{x+1})^2 = 9a^{10}x^2 + 30a^5x y^{x+1} + 25y^{2x+2}
\]

This is the expanded form of the given expression.

Would you like further details or any clarification?

### Related Questions:
1. What is the square of a binomial formula and how is it derived?
2. How do you expand powers of binomials for larger exponents?
3. What are some applications of binomial expansion in calculus?
4. Can this expression be further simplified or factored?
5. How would the expansion change if the exponents were different variables?

### Tip:
When expanding binomials, always start by identifying the individual terms and apply formulas systematically to avoid errors.

Explore how to expand the binomial expression (3a^5x + 5y^x+1)^2 using the square of a binomial formula. This step-by-step solution covers key algebraic concepts like exponents and binomial expansion, suitable for Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation