https://file.aiquickdraw.com/mathbot/file/17270185206386bd4t02i.jpg

The expression in the image is:

\[
\left( \frac{1}{3}a + \frac{2}{3}b \right)^3 - \left( \frac{1}{3}a - \frac{2}{3}b \right)^3
\]

We are tasked with simplifying this expression. This is a difference of cubes, and we can apply the identity for the difference of cubes:

\[
x^3 - y^3 = (x - y)(x^2 + xy + y^2)
\]

Let:

\[
x = \frac{1}{3}a + \frac{2}{3}b \quad \text{and} \quad y = \frac{1}{3}a - \frac{2}{3}b
\]

Thus, we can apply the difference of cubes formula:

1. Subtract the two terms in parentheses:

\[
x - y = \left( \frac{1}{3}a + \frac{2}{3}b \right) - \left( \frac{1}{3}a - \frac{2}{3}b \right)
\]

Simplifying:

\[
x - y = \frac{1}{3}a + \frac{2}{3}b - \frac{1}{3}a + \frac{2}{3}b = \frac{4}{3}b
\]

2. Compute \( x^2 + xy + y^2 \). We'll calculate each part separately.

- \( x^2 = \left( \frac{1}{3}a + \frac{2}{3}b \right)^2 = \frac{1}{9}a^2 + \frac{4}{9}ab + \frac{4}{9}b^2 \)
- \( y^2 = \left( \frac{1}{3}a - \frac{2}{3}b \right)^2 = \frac{1}{9}a^2 - \frac{4}{9}ab + \frac{4}{9}b^2 \)
- \( xy = \left( \frac{1}{3}a + \frac{2}{3}b \right) \left( \frac{1}{3}a - \frac{2}{3}b \right) = \frac{1}{9}a^2 - \frac{4}{9}b^2 \)

Adding these together:

\[
x^2 + xy + y^2 = \left( \frac{1}{9}a^2 + \frac{4}{9}ab + \frac{4}{9}b^2 \right) + \left( \frac{1}{9}a^2 - \frac{4}{9}b^2 \right) + \left( \frac{1}{9}a^2 - \frac{4}{9}ab + \frac{4}{9}b^2 \right)
\]

Simplifying:

\[
x^2 + xy + y^2 = \frac{3}{9}a^2 + \frac{8}{9}b^2
\]

Therefore, the fully simplified expression becomes:

\[
\frac{4}{3}b \times \left( \frac{3}{9}a^2 + \frac{8}{9}b^2 \right) = \frac{4}{3}b \times \frac{1}{3}(a^2 + 8b^2) = \frac{4}{9}b(a^2 + 8b^2)
\]

The final simplified form is:

\[
\frac{4}{9}b(a^2 + 8b^2)
\]

Would you like further details on any step?

Here are five related questions:
1. How would this formula change if the exponents were different (e.g., squared instead of cubed)?
2. What is the difference between a sum of cubes and a difference of cubes?
3. How can the binomial expansion help in expanding powers of sums or differences?
4. What happens if \(a = b\) in this particular expression?
5. How does factoring help simplify cubic expressions?

**Tip:** When simplifying expressions involving cubes, look for patterns such as the difference or sum of cubes to apply special identities.

Simplify the expression: (1/3 a + 2/3 b)^3 - (1/3 a - 2/3 b)^3.

Learn how to simplify the expression (1/3 a + 2/3 b)^3 - (1/3 a - 2/3 b)^3 using the difference of cubes identity. This step-by-step guide covers algebraic simplifications, key formulas, and suitable methods for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation