Problem 88 - 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: Combine the Exponents in the Numerator

The first step is to apply the rule of adding exponents when multiplying like bases: \(b^{m} \times b^{n} = b^{m+n}\). Here, we have the same base \(b\) in the numerator: \(b^{4/5} \times b^{4/5}\). Adding the exponents, we get: \(b^{4/5 + 4/5} = b^{8/5}\).

Step 2: Simplify the Entire Fraction

With the numerator now simplified to \(b^{8/5}\), apply the property of exponents that states \(\frac{b^{m}}{b^{n}} = b^{m-n}\). Implement this rule to the entire fraction: \(\frac{b^{8/5}}{b^{3/5}} = b^{8/5 - 3/5} = b^{5/5}\).

Step 3: Simplify Exponents

The exponent \(b^{5/5}\) can be simplified further since anything raised to the power of 1 equals itself: \(b^{5/5} = b^{1} = b\). So, the expression simplifies to \(b\).

Key Concepts

Laws of Exponents

Understanding the laws of exponents is crucial when simplifying expressions involving powers and variable bases. These rules help manage and reduce complex expressions without doing unnecessary calculations.
Here are some key laws:

• Multiplication of like bases: For any real number base and exponents, \(a^m \times a^n = a^{m+n}\). This rule states you can add exponents if the bases are the same.
• Division of like bases: When dividing, use \(a^m \div a^n = a^{m-n}\). Subtract the exponents if the bases match.
• Power to a power: Raising an exponent to another exponent is done by multiplying: \( (a^m)^n = a^{m \times n}\).
• Anything to the power of zero is one: \(a^0 = 1\), given that \(a\) is not zero.

Always keep these rules in mind while simplifying any expression with exponents, as the properties streamline the process.

Multiplying Expressions

Multiplying expressions involving exponents follows a straightforward principle, especially when working with the same base.
When you multiply expressions like \(b^{4/5} \times b^{4/5}\), you apply the multiplication rule: add the exponents if the bases are identical.
This is because the product rule for exponents essentially condenses repeated multiplication into a single expression.
For example:

• Start with the expression: \(b^{4/5} \times b^{4/5}\)
• Add the exponents due to the common base: \(4/5 + 4/5 = 8/5\)
• Result: \(b^{8/5}\)

These steps simplify the multiplication by replacing repetitive calculations with a single, more manageable form.

Fractional Exponents

Fractional exponents might seem intimidating, but they follow similar rules to whole number exponents. A fractional exponent represents both a root and a power.
The general form \(a^{m/n}\) means the\(n\)-th root of \(a^m\). For instance:

• In our case, \(b^{4/5}\) signifies the 5th root of \(b\) raised to the 4th power.

Working with fractional exponents also allows us to:

• Simplify combinations of roots and powers into a single expression.
• Convert expressions between radical and exponent form, which can be useful for different types of calculations.

By understanding fractional exponents, you gain flexibility in how you manage and simplify expressions, enabling you to handle more complex algebra easily.

Positive Exponents

Positive exponents make the expression straightforward by denoting repeated multiplication of a base. Using positive exponents is crucial for simplifying expressions and ensures that the result is in its simplest form.
Positive exponents indicate how many times a number is multiplied by itself. For instance:

• \(b^3\) means \(b\) multiplied by itself twice more: \(b \times b \times b\).
• During the simplification step of our problem, we aim to have all exponents positive, thereby making \(b^{5/5} = b^1 = b\).

Expanding an expression to positive exponents often means re-evaluating any negative exponents in the initial problem. This ensures clarity and correctness in the simplification process.