 Find the product.
left parenthesis x Superscript 7 Baseline y Superscript 4 Baseline minus 5 right parenthesis squared
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Part 1
left parenthesis x Superscript 7 Baseline y Superscript 4 Baseline minus 5 right parenthesis squaredequals
select: $$    (x^7 y^4 - 5)^2 = a^2 - 2ab + b^2 = x^{14} y^8 - 10x^7 y^4 + 25    $$
 ​(Simplify your​ answer.)


We are tasked with simplifying $(x^7 y^4 - 5)^2$. This expression can be expanded using the formula for the square of a binomial:

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

In this case, $a = x^7 y^4$ and $b = 5$.

### Step-by-step solution:

1. **Square the first term:**
   \[
   a^2 = (x^7 y^4)^2 = x^{14} y^8
   \]

2. **Multiply the first and second terms by 2:**
   \[
   2ab = 2(x^7 y^4)(5) = 10x^7 y^4
   \]

3. **Square the second term:**
   \[
   b^2 = 5^2 = 25
   \]

Now, combine all terms:

\[
(x^7 y^4 - 5)^2 = x^{14} y^8 - 10x^7 y^4 + 25
\]

This is the simplified expression.

Would you like any further details on this solution?

Here are five related questions to consider:
1. How do you expand the square of a binomial when the terms have exponents?
2. What is the general form of the binomial expansion for any $(a + b)^2$?
3. How does the distribution of exponents work when multiplying variables?
4. How would this expression change if it were $(x^7 y^4 + 5)^2$ instead?
5. How can you apply this binomial expansion method to more complex polynomials?

**Tip:** When expanding a binomial, always apply the distributive property step by step to avoid mistakes.

Find the product (x^7 y^4 - 5)^2. Simplify your answer.

Learn how to simplify the expression (x^7 y^4 - 5)^2 step by step using the binomial expansion formula. This problem covers key algebraic concepts such as exponents and binomial expansion. Suitable for students in Grades 9-11.

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