Problem 16 - Applied Calculus

Question

Evaluate each expression without using a calculator. $$ \left[\left(\frac{2}{5}\right)^{-2}\right]^{-1} $$

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{4}{25}\).

Step 1: Understand Negative Exponent

When you have a term like \(x^{-n}\), it represents the reciprocal of \(x\) raised to the positive power \(n\). Therefore, \(\left(\frac{2}{5}\right)^{-2}\) is the reciprocal of \(\left(\frac{2}{5}\right)^2\).

Step 2: Simplify Inner Expression

Now calculate \(\left(\frac{2}{5}\right)^2\), which equals \(\frac{2^2}{5^2} = \frac{4}{25}\). Thus, \(\left(\frac{2}{5}\right)^{-2}\) is the reciprocal of \(\frac{4}{25}\), which is \(\frac{25}{4}\).

Step 3: Simplify Outer Expression

Now apply the negative exponent to the entire expression: \(\left(\frac{25}{4}\right)^{-1}\). This represents the reciprocal of \(\frac{25}{4}\), which is \(\frac{4}{25}\).

Step 4: Review the Final Result

The entire original expression \(\left[\left(\frac{2}{5}\right)^{-2}\right]^{-1}\) simplifies to \(\frac{4}{25}\).

Key Concepts

Reciprocal

Reciprocal helps you understand the magic behind division and inverse operations. Imagine you are flipping a number upside down; that's essentially what finding the reciprocal means. For example, if you take a fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \).
Working with reciprocals is crucial when dealing with negative exponents. Notice that \( x^{-n} \) involves taking the reciprocal of \( x \) raised to the power \( n \). So, \( x^{-n} = \frac{1}{x^n} \).
Reciprocals are self-explanatory with whole numbers too: the reciprocal of 2 is \( \frac{1}{2} \). It's helpful to practice finding reciprocals because their applications are wide-ranging, especially in algebra and calculus.

Exponentiation

Exponentiation is the process of raising a quantity to a power. When we write \( x^n \), it means multiplying \( x \) by itself \( n \) times.
In the exercise, exponentiation is applied to a fraction, illustrating the power of exponents over both the numerator and denominator. For instance, \( \left(\frac{2}{5}\right)^2 \) expands to \( \frac{2^2}{5^2} \), which simplifies to \( \frac{4}{25} \).
The laws of exponents, like multiplying same bases (\( x^a \times x^b = x^{a+b} \)), also apply here. Understanding these rules helps simplify expressions and solve equations efficiently. Remember, when dealing with negative exponents, this concept ties directly into finding reciprocals.

Fraction

Fractions represent parts of a whole. They consist of a numerator and a denominator, separated by a line, like \( \frac{a}{b} \). The numerator tells how many parts you have, and the denominator tells the kind of parts you are dealing with.
In mathematics, fractions allow for dividing things evenly and understanding proportions. When you square a fraction, like \( \left(\frac{2}{5}\right)^2 \), you square both the top and the bottom, resulting in \( \frac{4}{25} \).
When working on solutions involving fractions, it is important to know operations like addition, subtraction, multiplication, and division. Familiarity with converting improper fractions and mixed numbers enhances fluency in solving complex problems and is essential in many areas of math.