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Chapter 10: Problem 354

In the following exercises, simplify. $$a^{3} b^{-3}$$

Short Answer

Expert verified
The simplified form is \(\frac{a^{3}}{b^{3}}\).

Step by step solution

01

Understand the Problem

The goal is to simplify the expression \(a^{3} b^{-3}\) using the rules of exponents.
02

Apply the Rule of Negative Exponents

A negative exponent indicates that the base should be moved to the opposite side of the fraction line (numerator to denominator or vice versa) and the exponent made positive. Hence, \(b^{-3}\) becomes \(\frac{1}{b^{3}}\).
03

Rewrite the Expression

Rewrite \(a^{3} b^{-3}\) using the result from the previous step. It becomes \(a^{3} \cdot \frac{1}{b^{3}}\).
04

Simplify the Expression

Combine the terms to get a single fraction: \(\frac{a^{3}}{b^{3}}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Negative Exponents
A negative exponent indicates that the base is on the wrong side of the fraction line and needs to be moved to the opposite side.
For instance, if you have a term like \(b^{-3}\), this can be rewritten as \( \frac{1}{b^{3}} \).
If the negative exponent is in the denominator like \( \frac{1}{b^{-3}} \), you move it to the numerator to remove the negative sign, making it \(b^{3}\).
This rule helps in simplifying expressions by eliminating negative exponents.
So in our example, \(a^{3} b^{-3}\) becomes \(a^{3} \cdot \frac{1}{b^{3}}\).
Rules of Exponents
The rules of exponents help us simplify complex expressions involving powers.
Here are some essential ones:
  • Product of Powers: \(a^m \cdot a^n = a^{m+n}\)
    When you multiply powers with the same base, add the exponents.
  • Quotient of Powers: \( \frac{a^m}{a^n} = a^{m-n} \)
    When you divide powers with the same base, subtract the exponents.
  • Power of a Power: \((a^m)^n = a^{m \cdot n}\)
    When raising a power to another power, multiply the exponents.
  • Negative Exponents: \(a^{-m} = \frac{1}{a^m}\)
    Move the base to the opposite side of the fraction line and make the exponent positive.
  • Zero Exponent: \(a^0 = 1\)
    Any base with an exponent of zero equals one.
In our problem, understanding the negative exponents and product of powers rules is critical for simplifying step-by-step.
Fraction Simplification
Simplifying fractions involves reducing them to the simplest form.
Here's how you can achieve this:
  • Common Factors: Identify if the numerator and denominator have common factors and divide both by the greatest common factor (GCF).
    For example, the fraction \( \frac{12}{16} \) can be simplified by dividing both by their GCF, which is 4, to get \( \frac{3}{4} \).
  • Combining Rates: If you have a fraction involving powers, simplify using the rules of exponents.
    For \( \frac{a^{3}}{b^{3}} \), check if there's any further reduction possible. Here, since the bases \(a\) and \(b\) are different, we leave the expression as is.
Understand these steps thoroughly to make fraction simplification easier!

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