Problem 60 - Intermediate Algebra

Question

Use the properties of exponents to simplify each expression. Write with positive exponents. $$ \frac{\left(a^{-2} b^{3}\right)^{1 / 8}}{\left(a^{-3} b\right)^{-1 / 4}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{b^{5/8}}{a} \).

Step 1: Apply the Power of a Power Rule

Begin by applying the power of a power rule, \[(x^m)^n = x^{mn}\] to each part of the expression. For the numerator, we have \[(a^{-2}b^3)^{1/8} = a^{-2\cdot\frac{1}{8}}b^{3\cdot\frac{1}{8}} = a^{-\frac{1}{4}}b^{\frac{3}{8}}\]For the denominator, \[(a^{-3}b)^{-1/4} = a^{-3\cdot(-\frac{1}{4})}b^{-\frac{1}{4}} = a^{\frac{3}{4}}b^{-\frac{1}{4}}\]

Step 2: Rewrite the Expression as a Division

The expression can be rewritten replacing the simplified numerator and denominator:\[ \frac{a^{-\frac{1}{4}}b^{\frac{3}{8}}}{a^{\frac{3}{4}}b^{-\frac{1}{4}}} \]

Step 3: Apply Laws of Exponents for Division

Use the law of exponents for division: \[ \frac{x^a}{x^b} = x^{a-b} \]Apply this to both the \(a\) and \(b\) terms:\[ a^{-\frac{1}{4} - \frac{3}{4}}, \quad b^{\frac{3}{8} - (-\frac{1}{4})} \]This gives us:\[ a^{-1}, \quad b^{\frac{3}{8} + \frac{1}{4}} = b^{\frac{3}{8} + \frac{2}{8}} = b^{\frac{5}{8}} \]

Step 4: Write with Positive Exponents

The simplified expression is a negative exponent indicates a reciprocal, so re-write any negative exponents as positive:\[ \frac{b^{\frac{5}{8}}}{a} \]

Key Concepts

Power of a Power Rule

The power of a power rule is a fundamental aspect of working with exponents. This rule states that when you have an exponent raised to another exponent, you multiply the exponents together. Mathematically, this is expressed as: \[(x^m)^n = x^{m \cdot n}\] This property allows you to simplify expressions where terms with exponents have further exponents applied to them.

• Consider the expression \((a^{-2}b^3)^{1/8}\).
• Using the power of a power rule, \(a^{-2}\) raised to \(1/8\) becomes \(a^{-2 \cdot \frac{1}{8}} = a^{-\frac{1}{4}}\).
• Similarly, \(b^3\) raised to \(1/8\) becomes \(b^{3 \cdot \frac{1}{8}} = b^{\frac{3}{8}}\).

This simplification helps in handling more complex expressions and reduces them into more manageable pieces.

Laws of Exponents for Division

Division involving exponents often leads to confusion, but understanding the laws can make it simpler. When you divide like bases with exponents, you subtract the exponent of the denominator from the exponent of the numerator. The law can be expressed as: \[ \frac{x^a}{x^b} = x^{a-b} \] This means that dividing two powers with the same base reduces to simply subtracting their exponents.

• Taking our example with \(a^{\text{exponents}}\), \(\frac{a^{-\frac{1}{4}}}{a^{\frac{3}{4}}}\), you apply the law and get \(a^{-\frac{1}{4} - \frac{3}{4}} = a^{-1}\).
• For \(b^{\text{exponents}}\), \(\frac{b^{\frac{3}{8}}}{b^{-\frac{1}{4}}}\), you subtract and find \(b^{\frac{3}{8} + \frac{1}{4}} = b^{\frac{5}{8}}\).

Understanding and applying these rules prevents you from making common errors associated with exponents during division.

Simplifying Expressions with Exponents

Simplifying expressions with exponents involves converting complex expressions into a form that is easier to understand and work with. The main goal is to express all parts of the expression using positive exponents and simplified terms. Let’s see how this works in practice. Start by simplifying each component using the rules we've discussed.

• We rewritten our original expression as \(\frac{a^{-1}b^{\frac{5}{8}}}{a}\), which has negative and positive exponents.
• A negative exponent, like \(a^{-1}\), indicates the reciprocal, so \(a^{-1} = \frac{1}{a}\).

Therefore, the entire expression becomes \(\frac{b^{\frac{5}{8}}}{a}\), now fully simplified using only positive exponents. This makes the expression much neater and easier to interpret or use in further calculations.