Problem 12 - Algebra for College Students

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 Reciprocal Property

The expression involves a fraction raised to a negative power. Recall that a number to a negative power is the reciprocal of the number to the positive power. Therefore, for any non-zero number a: \[ a^{-n} = \frac{1}{a^n} \]In this case, the expression has \( \left( \frac{4}{5} \right)^{-2} \). By applying the reciprocal property, this becomes the reciprocal of \( \left( \frac{4}{5} \right)^{2} \).

Step 2: Apply the Negative Exponent Rule

Following the rule from step 1, if an expression is in the form \( \frac{1}{a^{-n}} \), it simplifies directly to \( a^{n} \). Here, the expression is \( \frac{1}{\left(\frac{4}{5}\right)^{-2}} \), so this simplifies directly to \( \left(\frac{4}{5}\right)^{2} \).

Step 3: Calculate the Square

Now, calculate \( \left( \frac{4}{5} \right)^{2} \). Squaring a fraction means squaring both the numerator and the denominator:\[ \left( \frac{4}{5} \right)^{2} = \frac{4^2}{5^2} = \frac{16}{25} \]

Step 4: Final Simplified Expression

The expression \( \frac{1}{\left(\frac{4}{5}\right)^{-2}} \) simplifies to \( \frac{16}{25} \). Thus, your simplified expression is complete.

Key Concepts

Reciprocal Property

When working with expressions that involve negative exponents, understanding the reciprocal property is crucial. The reciprocal of a number is essentially 'flipping' it, or switching the numerator and the denominator. For example, the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \).

The reciprocal property is tied closely to the concept of negative exponents. They tell us the expression is upside down and needs to be flipped. A negative exponent like \( a^{-n} \) indicates that instead of multiplying by \( a \), we are dividing by its positive power. This leads to the formula:

• \( a^{-n} = \frac{1}{a^n} \)

When you see an expression like \( \left( \frac{4}{5} \right)^{-2} \), apply the reciprocal property by turning the base fraction into its reciprocal raised to the positive power. Understand it as decreasing power and flipping order to get a positive exponent, leading us to evaluate using the positive power instead.

Negative Exponent Rule

The negative exponent rule helps us transform expressions into more workable forms. It essentially means that the negative exponent signals that the base (number or fraction) needs its position changed from numerator to denominator or vice versa.

Whenever faced with a negative exponent, you can convert it to a positive exponent by using the reciprocal as the first step in simplification. For instance, if you have \( \frac{1}{a^{-n}} \), it changes directly to \( a^{n} \). This conversion simplifies our computations considerably.

In our case, \( \frac{1}{\left( \frac{4}{5} \right)^{-2}} \) simplifies immediately to \( \left( \frac{4}{5} \right)^{2} \). This move bypasses the need to deal with negative exponents outright and allows us to continue with more straightforward computations.

Squaring Fractions

Squaring fractions is a straightforward process once you understand the basics of working with fractions. Squaring means to multiply a number by itself. When you square a fraction, you need to apply the squaring separately to both the numerator and the denominator.

Suppose you have a fraction \( \frac{a}{b} \). To find \( \left( \frac{a}{b} \right)^2 \) (square it), follow these steps:

• Square the numerator: \( a^2 \).
• Square the denominator: \( b^2 \).
• Form the fraction: \( \frac{a^2}{b^2} \).

This approach maintains equal operations on both parts of the fraction and results in a new, simplified form of the expression. In our exercise, we computed \( \left( \frac{4}{5} \right)^2 \) by squaring 4 to get 16 and 5 to get 25, giving us the simplified \( \frac{16}{25} \).