Problem 75 - Elementary and Intermediate Algebra

Question

Use rules for exponents to simplify each expression. $$ \frac{\left(a b^{2}\right)^{3}}{a^{2} b^{2}} $$

Step-by-Step Solution

Answer

The simplified expression is \( ab^4 \).

Step 1: Exponentiation Inside the Numerator

Start by considering the expression inside the numerator, \( (ab^2)^3 \). To simplify, use the power of a product rule: \( (xy)^n = x^n y^n \). This gives \( a^3 (b^2)^3 \).

Step 2: Simplifying with Power of a Power Rule

Use the power of a power rule: \( (b^m)^n = b^{mn} \) to simplify \( (b^2)^3 \). This equals \( b^{2 imes 3} = b^6 \). Now the expression becomes \( a^3 b^6 \) in the numerator.

Step 3: Write Complete Fraction

Return to the original fraction with the simplified numerator. It now reads: \( \frac{a^3 b^6}{a^2 b^2} \).

Step 4: Apply Quotient Rule for Exponents

Use the quotient rule where \( \frac{x^m}{x^n} = x^{m-n} \) for simplification. Simplify each base separately: 1. For \( a \), \( \frac{a^3}{a^2} = a^{3-2} = a^1 \).2. For \( b \), \( \frac{b^6}{b^2} = b^{6-2} = b^4 \).

Step 5: Final Simplification

Combine the simplified forms of \( a \) and \( b \). The expression now is \( ab^4 \).

Key Concepts

Power of a Product Rule

The power of a product rule is a handy tool when you want to raise a product of two numbers or variables to a power. It's pretty straightforward once you get the hang of it.
When you see an expression like

• \((ab^2)^3\), you use the rule that says \((xy)^n = x^n y^n\).

This means that each factor inside the parentheses should be raised to the given power separately. In this exercise, we took \( (ab^2)^3 \) and transformed it into \( a^3 (b^2)^3 \). So, instead of multiplying three copies of \( ab^2 \), we simplify by handling each part independently.
It's like keeping ingredients separate while baking, ensuring uniformity before mixing. This approach simplifies complex expressions greatly and is essential in maneuveing through exponent rules effectively.

Power of a Power Rule

The power of a power rule is specifically used when you have an exponent raised to another exponent, such as in the expression \((b^2)^3\).
According to this rule, when you have \((x^m)^n\), it becomes \(x^{m \times n}\).
In our problem, the second step involved converting \((b^2)^3\) into \(b^{2 \times 3} = b^6\). This means you're effectively multiplying the exponents together.
This principle helps simplify deeper nested exponentiation and saves time by turning multi-step multiplication into a simple exponent equation. It's like having a shortcut on a journey, making complex routes more straightforward. As you practice, this rule will become a neat trick up your sleeve for working with exponential expressions efficiently.

Quotient Rule for Exponents

The quotient rule for exponents helps simplify expressions where one exponent is divided by another. It's especially useful when dealing with fractions. The rule says that \(\frac{x^m}{x^n} = x^{m-n}\).
In our problem, the expression \(\frac{a^3 b^6}{a^2 b^2}\) was present.

To simplify:

• For the variable \(a\), the approach followed was \(\frac{a^3}{a^2} = a^{3-2} = a^1\).
• For the variable \(b\), it was \(\frac{b^6}{b^2} = b^{6-2} = b^4\).

This technique allows the base to remain while you subtract the exponents of like bases, making it easier to handle division in equations.
Understanding and applying the quotient rule streamlines the process of simplifying fractions in algebra significantly.