Problem 80 - Elementary and Intermediate Algebra

Question

Simplify each expression. Write the answers without negative exponents. All variables represent positive real numbers. See Example 8. $$ 5^{1 / 3} 5^{-5 / 3} $$

Step-by-Step Solution

Answer

\(\frac{1}{5^{4/3}}\)

Step 1: Apply the Product of Powers Rule

When multiplying powers with the same base, we add the exponents. The expression is \(5^{1/3} \cdot 5^{-5/3}\). Add the exponents: \(\frac{1}{3} + \left(-\frac{5}{3}\right) = \frac{1}{3} - \frac{5}{3}\).

Step 2: Simplify the Exponent

Simplify the expression \(\frac{1}{3} - \frac{5}{3}\) by finding a common denominator. Both fractions have the same denominator, so subtract the numerators: \(\frac{1 - 5}{3} = \frac{-4}{3}\).

Step 3: Express With Positive Exponents

The result from the previous step is \(\frac{-4}{3}\), which is a negative exponent. To express it with a positive exponent, use the rule \(a^{-b} = \frac{1}{a^b}\). Thus, \(5^{-4/3} = \frac{1}{5^{4/3}}\).

Key Concepts

Product of Powers Rule

The Product of Powers Rule is a fundamental concept in algebra that helps simplify expressions involving exponents. When you multiply two powers that have the same base, you can simply add their exponents. This can make solving exponent-related problems much more manageable. For example:

• If you have the expression \( a^m \cdot a^n \), where both terms have the base \( a \), you can apply the rule: \( a^{m+n} \).
• This means for \( 5^{\frac{1}{3}} \cdot 5^{-\frac{5}{3}} \), since both have the base 5, you add the exponents.

This rule simplifies the process of handling multiple exponents, as it reduces the number of steps necessary to reach the solution. However, ensure that only bases that are the same can use this rule to avoid mistakes.

Negative Exponents

Understanding negative exponents is crucial because they often appear unexpectedly in algebraic operations. A negative exponent suggests that you have a reciprocal or an "inverted" power.

• The rule for negative exponents tells us that \( a^{-b} \) is the same as \( \frac{1}{a^b} \).
• In practice, this means if you have \( 5^{-\frac{4}{3}} \), you can express it as \( \frac{1}{5^{\frac{4}{3}}} \).

Negative exponents indicate division into smaller fractions. By converting them into positive exponents, we often make expressions easier to understand and compare. Remember, negative exponents don't make numbers negative but rather smaller fractions.

Simplifying Exponents

Simplifying exponents is an essential skill for making expressions easy to read and solve. This involves reducing complex expressions to their simplest form while ensuring accuracy. To do this, follow these steps:

• A common strategy is to always seek to combine like terms, using rules like the Product of Powers Rule.
• Next, if you encounter negative exponents, convert them into positive exponents by finding the reciprocal.

For instance, transforming \( 5^{1/3} \cdot 5^{-5/3} \) by applying the addition of exponents results in \( 5^{-4/3} \). Then, rewriting \( 5^{-4/3} \) as \( \frac{1}{5^{4/3}} \) further simplifies the expression by removing the negative exponent.These steps make sure you understand every part of the process, providing clarity to often confusing algebraic expressions. Simplification is about making expressions as straightforward and clear as possible while complying with mathematical rules.