Problem 88 - Elementary and Intermediate Algebra

Question

Simplify each expression. Write the answers without negative exponents. All variables represent positive real numbers. See Example 8. $$ \frac{b^{4 / 5} b^{4 / 5}}{b^{3 / 5}} $$

Step-by-Step Solution

Answer

The simplified expression is \( b \).

Step 1: Multiply the Numerator Exponents

First, multiply the exponents in the numerator: \( b^{4/5} \times b^{4/5} \). When multiplying like bases, we add the exponents: \[b^{4/5 + 4/5} = b^{8/5}.\]

Step 2: Simplify the Expression

With the simplified numerator \( b^{8/5} \), your expression becomes: \[\frac{b^{8/5}}{b^{3/5}}.\]

Step 3: Subtract the Exponents in the Fraction

To simplify the fraction, subtract the exponent in the denominator from the exponent in the numerator: \[b^{8/5 - 3/5} = b^{5/5} = b^1.\]

Step 4: Remove Unnecessary Exponents

Any base to the power of 1 is simply the base itself: \[b^1 = b.\]

Key Concepts

Multiplying Exponents

When multiplying exponents with the same base, remember that it is much simpler than it looks. You simply need to add the exponents together. This rule is crucial and helps in preventing mistakes.
If you have an expression like: - \( a^m \times a^n \), where \(a\) is the base and \(m\) and \(n\) are the exponents, you can combine them by adding the exponents: - \( a^{m+n} \).
This rule makes handling expressions like \( b^{4/5} \times b^{4/5} \) straightforward. By adding \( 4/5 + 4/5 \), you get \( 8/5 \). As a result, the product of the exponents becomes \( b^{8/5} \).
Always make sure that the bases you are dealing with are the same. If they aren't, this rule doesn't apply. When they're matched, the multiplication of powers becomes as simple as addition.

Adding Exponents

Adding exponents is a part of operations like multiplying them. However, it's not a direct operation you perform on standalone exponents. In mathematics, adding exponents means adding their values during multiplication of like bases.

To break it down: if you see \( b^m \times b^n \), the exponent addition occurs because you're actually multiplying the base \(b\) added each time. This looks like \( b \times b \times b \)...a total of \( m+n \) times for our example.

• Recognize: \( b^{m+n} \) is the outcome of multiplication.
• Observe: The operation simplifies and combines repeated multiplication into a single power.

This means multiplication of the same base harnesses the addition of exponents. So, in practical terms, when you're asked to multiply and thereby add \( b^{4/5+4/5} \), you reach \( b^{8/5} \).
Adding exponents via multiplication of bases builds a simpler yet effective expression.

Subtracting Exponents

Subtracting exponents is what you do when you divide like bases. It's an essential process enabling simplification of exponential expressions. Let's dig into this method.

When you have an expression such as \( \frac{b^m}{b^n} \): - The bases are the same, \(b\).
- The operation is to subtract the exponents: \( m - n \). This results in: - \( b^{m-n} \).
Say you're working with \( \frac{b^{8/5}}{b^{3/5}} \), you subtract: - \( 8/5 - 3/5 \), leaving \( 5/5 \), which simplifies to \( b^1 \).
Practically, subtraction here turns division of like bases into a manageable power expression. Once simplified to \( b^1 \), you recognize that any base to the power of one, like \( b \), just remains as itself.
Subtracting exponents, therefore, ensures expressions are concise and streamlined.