Problem 43 - Applied Calculus

Question

Evaluate each expression without using a calculator. $$ \left(\frac{25}{16}\right)^{-1 / 2} $$

Step-by-Step Solution

Answer

The expression evaluates to \( \frac{4}{5} \).

Step 1: Understand the expression

The expression we need to evaluate is \( \left( \frac{25}{16} \right)^{-1/2} \). This means we need to find the reciprocal of the square root of \( \frac{25}{16} \).

Step 2: Calculate the square root of the fraction

First, find the square root of the numerator and the denominator separately. The square root of \( 25 \) is \( 5 \), and the square root of \( 16 \) is \( 4 \). Thus, \( \sqrt{\frac{25}{16}} = \frac{\sqrt{25}}{\sqrt{16}} = \frac{5}{4} \).

Step 3: Apply the negative exponent

Since the expression has a negative exponent, we find the reciprocal of \( \frac{5}{4} \). The reciprocal of \( \frac{5}{4} \) is \( \frac{4}{5} \).

Step 4: Combine the results

By combining these steps, \( \left( \frac{25}{16} \right)^{-1/2} = \frac{4}{5} \).

Key Concepts

Understanding Negative Exponents

Negative exponents might initially seem intimidating, but they are fairly simple when broken down. A negative exponent essentially indicates that you are taking the reciprocal of the base raised to the corresponding positive exponent.
For instance, consider a number raised to the power of \(-n\), written as \(a^{-n}\). This can be rewritten as \(\frac{1}{a^n}\). By doing this, you are effectively flipping or inverting the base.
This concept is useful in various mathematical operations, including the simplification of expressions and solving complex equations. Remember, a negative exponent doesn't mean the result is a negative number; rather, it signifies the turning of the number upside-down in terms of its place on the number line.

• **Example:** \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\)
• **Concept:** Flip the base when dealing with negative exponents.

Exploring Square Roots

Square roots are an integral part of many areas in mathematics. Taking the square root of a number involves finding another number that, when multiplied by itself, yields the original number.
For example, the square root of \(25\) is \(5\), because \(5 \times 5 = 25\). Similarly, the square root of \(16\) is \(4\) since \(4 \times 4 = 16\).
When dealing with fractions, you simply find the square roots of both the numerator and the denominator individually. Thus, for the fraction \(\frac{25}{16}\), you calculate as follows: \(\sqrt{\frac{25}{16}} = \frac{\sqrt{25}}{\sqrt{16}} = \frac{5}{4}\).

• **Method:** Square root of \(\frac{a}{b} = \frac{\sqrt{a}}{\sqrt{b}}\)
• **Practice:** Find the square root of each separately.

The Reciprocal of Fractions

Finding the reciprocal of a fraction is a straightforward process that involves swapping the numerator and the denominator. The reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\). It's like flipping the fraction upside down.
This operation is crucial when dealing with complex fractions, dividing fractions, and working with negative exponents. It helps transform mathematical expressions into simpler forms, making them easier to compute.
For example, if you have \(\frac{5}{4}\), its reciprocal is \(\frac{4}{5}\). This concept is often used in conjunction with other operations, such as in the exercise where \(\left(\frac{25}{16}\right)^{-1/2}\) was solved to \(\frac{4}{5}\) by applying the reciprocal.

• **Concept:** Swap the top and bottom numbers.
• **Application:** Useful in division and negative exponents.