Expanding Binomial Expressions with Exponents

Q13. Find the product: $\left(x^2 - 4y^3\right)^2$

Background

Topic: Binomial Expansion / Exponents

This question is testing your ability to expand a binomial expression raised to a power, specifically squaring a binomial. This is a fundamental skill in algebra, often used to simplify expressions and solve equations.

Key Terms and Formulas

• Binomial: An algebraic expression with two terms (e.g., $x^2 - 4y^3$).

• Exponent: Indicates how many times to multiply the expression by itself.

• Binomial Square Formula:

$\left(a - b\right)^2 = a^2 - 2ab + b^2$

Step-by-Step Guidance

1. Identify $a$ and $b$ in the expression $\left(x^2 - 4y^3\right)^2$. Here, $a = x^2$ and $b = 4y^3$.

2. Apply the binomial square formula: $\left(a - b\right)^2 = a^2 - 2ab + b^2$.

3. Calculate $a^2$: $\left(x^2\right)^2 = x^{2 \times 2} = x^4$.

4. Calculate $-2ab$: $-2 \times x^2 \times 4y^3 = -8x^2y^3$.

5. Calculate $b^2$: $\left(4y^3\right)^2 = 16y^{3 \times 2} = 16y^6$.

Try solving on your own before revealing the answer!

Final Answer: $x^4 - 8x^2y^3 + 16y^6$

We used the binomial square formula and substituted $a = x^2$ and $b = 4y^3$ to expand the expression.