Problem 12 - Intermediate Algebra

Question

Simplify each numerical expression. \(\frac{1}{\left(\frac{4}{5}\right)^{-2}}\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{16}{25} \).

Step 1: Understand the Negative Exponent

The negative exponent \(\left(\frac{4}{5}\right)^{-2}\) means you take the reciprocal of \(\frac{4}{5}\) and then raise it to the positive of that exponent. So, \(\left(\frac{4}{5}\right)^{-2}\) is equivalent to \(\left(\frac{5}{4}\right)^{2}\).

Step 2: Take the Reciprocal

Switch the numerator and denominator of \(\frac{4}{5}\) to get the reciprocal, which is \(\frac{5}{4}\).

Step 3: Apply the Positive Exponent

Now raise \(\frac{5}{4}\) to the power of 2. This means multiplying \(\frac{5}{4}\) by itself: \(\left(\frac{5}{4}\right)\times\left(\frac{5}{4}\right)=\frac{25}{16}\).

Step 4: Simplify the Main Expression

Replace \(\left(\frac{4}{5}\right)^{-2}\) with \(\frac{25}{16}\) in the original expression. So, \(\frac{1}{(\frac{25}{16})}\) can be simplified by taking the reciprocal of \(\frac{25}{16}\), resulting \(\frac{16}{25}\).

Key Concepts

Negative Exponents

Negative exponents can be confusing, but they're just another way to express division. A negative exponent tells you how many times to divide, rather than multiply, a number by itself. For example, a negative exponent like \(a^{-n}\) means you take 1 divided by \(a^n\). In simpler terms, \(a^{-n} = \frac{1}{a^n}\).

Let's apply this to fractions. If we have \((\frac{4}{5})^{-2}\), this negative exponent signals a two-step process. First, flip the fraction, and second, raise it to the power of positive 2.

Reciprocal

Understanding reciprocals can make fraction calculations much simpler. The reciprocal of a number is just the 'flipped' version of it.

For a fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\). When dealing with fractions raised to negative exponents, you first convert them to their reciprocals.

So, for \(\left(\frac{4}{5}\right)^{-2}\), the reciprocal becomes \(\frac{5}{4}\), allowing the exponent to be positive and easier to manage.

Fractions

Fractions represent a part of a whole and are a simple way to express division. A fraction is composed of a numerator and a denominator. The numerator is the top number, and the denominator is the bottom one.

When you simplify expressions with fractions, you often encounter operations like finding a common denominator or converting complex fractions into simpler ones. In this exercise, when simplifying the given fraction, we use the reciprocal, which turns the division into a multiplication.

Power of a Fraction

Raising a fraction to a power means multiplying the fraction by itself as many times as the power indicates. With positive exponents, this simply involves repeated multiplication. So \(\left(\frac{5}{4}\right)^2\) means \(\left(\frac{5}{4}\right) \times \left(\frac{5}{4}\right)\).

This results in \(\frac{25}{16}\), as you multiply the numerators and the denominators separately: \(5 \times 5 = 25\) and \(4 \times 4 = 16\).

Understanding the process of raising a fraction to a power can make complex expressions easier to handle.