Let's simplify the expression:

\[
\frac{{a^{-1} b^{-1}}}{{a^{-3} - b^{-3}}}
\]

### Step 1: Simplifying the numerator \((a^{-1} b^{-1})\)

Recall that:
\[
a^{-1} b^{-1} = \frac{1}{a} \cdot \frac{1}{b} = \frac{1}{ab}
\]

So the numerator becomes:
\[
\frac{1}{ab}
\]

### Step 2: Simplifying the denominator \((a^{-3} - b^{-3})\)

For negative exponents, recall that:
\[
a^{-3} = \frac{1}{a^3}, \quad b^{-3} = \frac{1}{b^3}
\]
Thus, the denominator becomes:
\[
\frac{1}{a^3} - \frac{1}{b^3}
\]

### Step 3: Finding the common denominator for \(\frac{1}{a^3} - \frac{1}{b^3}\)

The common denominator between \(a^3\) and \(b^3\) is \(a^3 b^3\). So we rewrite the expression:
\[
\frac{1}{a^3} - \frac{1}{b^3} = \frac{b^3 - a^3}{a^3 b^3}
\]

### Step 4: Substituting back into the original expression

Now, we substitute both the simplified numerator and denominator:
\[
\frac{\frac{1}{ab}}{\frac{b^3 - a^3}{a^3 b^3}} = \frac{1}{ab} \cdot \frac{a^3 b^3}{b^3 - a^3}
\]

### Step 5: Simplifying the expression

Multiply the numerators and denominators:
\[
= \frac{a^3 b^3}{ab (b^3 - a^3)} = \frac{a^2 b^2}{b^3 - a^3}
\]

Thus, the simplified form of the expression is:
\[
\frac{a^2 b^2}{b^3 - a^3}
\]

Would you like further details or have any questions?

Here are 5 related questions to explore:
1. What are the properties of negative exponents?
2. How do you simplify more complex fractional expressions with multiple variables?
3. Can you generalize this process for other powers of \(a\) and \(b\)?
4. What happens if \(a = b\) in the original expression?
5. How do you factor terms like \(b^3 - a^3\) in the denominator?

**Tip:** Remember that negative exponents indicate reciprocal terms, so converting them first often makes expressions easier to handle.

Learn how to simplify the algebraic expression (a^-1b^-1) / [(a^-3)-(b^-3)] step by step. This problem involves the use of negative exponents, fraction simplification, and finding a common denominator. Suitable 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