Problem 52 - Algebra and Trigonometry

Question

Simplify the expression and eliminate any negative exponents \((\mathrm{s}) .\) Assume that all letters denote positive numbers. \(\left(a^{2 / 5}\right)^{-3 / 4}\)

Step-by-Step Solution

Answer

\(\frac{1}{a^{3/10}}\) is the simplified expression.

Step 1: Understand the expression

The given expression is \( \left(a^{2/5}\right)^{-3/4} \). Our goal is to simplify this expression and eliminate any negative exponents. The expression includes a fractional exponent inside with another fractional exponent outside.

Step 2: Apply the Power of a Power Rule

First, recall the power of a power rule: \((x^m)^n = x^{m \times n}\). We apply this rule to the expression \(\left(a^{2/5}\right)^{-3/4}\). Multiply the exponents: \(a^{2/5 \cdot -3/4}\).

Step 3: Multiply the Exponents

Perform the multiplication of the exponents: \(\frac{2}{5} \times -\frac{3}{4} = -\frac{6}{20}\). This simplifies to \(-\frac{3}{10}\). The expression now is \(a^{-3/10}\).

Step 4: Eliminate the Negative Exponent

To eliminate the negative exponent, use the rule \(x^{-m} = \frac{1}{x^m}\). Thus, \(a^{-3/10}\) becomes \(\frac{1}{a^{3/10}}\). This expression has no negative exponents.

Key Concepts

Fractional Exponents

Fractional exponents, also known as rational exponents, are another way to represent roots. Instead of using the radical sign, we use the fraction to indicate the root operation.
For example, the expression \( x^{1/2} \) is equivalent to \( \sqrt{x} \), the square root of \( x \). Here, the numerator indicates the power, and the denominator indicates the root.
Another example could be \( x^{3/4} \), which represents \( (\sqrt[4]{x})^3 \), meaning you first take the fourth root of \( x \) and then cube it.
Understanding how fractional exponents work is important because it allows for alternative representations of expressions, which can make algebraic manipulation easier. This concept is especially useful when simplifying complex expressions with multiple layers of exponents.

Negative Exponents

Negative exponents indicate a reciprocal. When you see a negative exponent, think of it as moving the base to the opposite part of a fraction.
For instance, \( x^{-m} = \frac{1}{x^m} \). Here, \( x \) is moved from the numerator to the denominator, and the exponent sign is changed from negative to positive.
This is particularly useful for simplifying expressions and eliminating negative exponents by converting them into fractions.

• Example 1: \( 2^{-3} = \frac{1}{2^3} = \frac{1}{8} \)
• Example 2: \( y^{-1/2} = \frac{1}{y^{1/2}} = \frac{1}{\sqrt{y}} \)

By transforming negative exponents, you ensure expressions are more straightforward and often easier to work with, as demonstrated in the original solution when simplifying \( a^{-3/10} \) to \( \frac{1}{a^{3/10}} \).

Power of a Power Rule

The power of a power rule is a fundamental concept in exponents. When you have an expression like \((x^m)^n\), you can simplify it by multiplying the exponents: \( x^{m \times n} \).
This rule helps to manage multiple layers of exponents by condensing them into a single power, which can significantly simplify complex algebraic expressions.

• Example 1: \((z^2)^3 = z^{2 \times 3} = z^6\)
• Example 2: \((b^{5/6})^{3/5} = b^{5/6 \times 3/5} = b^{1/2}\)

Understanding this rule allows you to simplify expressions confidently and efficiently, as shown when transforming the original expression \( \left(a^{2/5}\right)^{-3/4} \) to \( a^{2/5 \times -3/4} = a^{-3/10} \). Recognizing when and how to apply the power of a power rule is crucial for reducing the complexity of expressions.