Problem 43 - Brief Applied Calculus

Question

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

Step-by-Step Solution

Answer

The value of the expression is \( \frac{4}{5} \).

Step 1: Understand the Negative Exponent

The expression \( \left( \frac{25}{16} \right)^{-1 / 2} \) involves a negative exponent. A negative exponent indicates the reciprocal of the base raised to the positive of that exponent. This means we need to take the reciprocal of \( \frac{25}{16} \) and raise it to the \( \frac{1}{2} \) power.

Step 2: Calculate the Reciprocal

Taking the reciprocal of \( \frac{25}{16} \) gives us \( \frac{16}{25} \). Now the expression becomes \( \left( \frac{16}{25} \right)^{1/2} \).

Step 3: Evaluate the Square Root

The \( \frac{1}{2} \) exponent indicates taking the square root of the base. So, we need to find \( \sqrt{\frac{16}{25}} \).

Step 4: Simplify the Square Root

The square root of a fraction \( \frac{a}{b} \) is \( \frac{\sqrt{a}}{\sqrt{b}} \). Calculate \( \sqrt{16} = 4 \) and \( \sqrt{25} = 5 \). Therefore, \( \sqrt{\frac{16}{25}} = \frac{4}{5} \).

Key Concepts

Negative Exponents

Negative exponents might seem tricky at first, but they're not too bad once you get the hang of it. A negative exponent indicates that you need to take the reciprocal of the base. But what is a reciprocal? Let's explore.

• If you see a negative exponent, like \[x^{-1}\], switch it to its reciprocal form \[\frac{1}{x}\].
• This means you are essentially "flipping" the base fraction over.
• For example, in the expression \[\left( \frac{25}{16} \right)^{-1/2}\], start by taking the reciprocal so that it becomes \[\left( \frac{16}{25} \right)^{1/2}\].

The negative exponent step is crucial as it sets the path for simplifying the expression correctly without a calculator.

Reciprocal

The reciprocal is essentially a mathematical term for "opposite" in terms of position. When you flip a fraction, you get its reciprocal. Let's break it down.

• The reciprocal of a number \(a\) is simply \(\frac{1}{a}\).
• For fractions, you invert the numerator and the denominator. For instance, the reciprocal of \(\frac{25}{16}\) is \(\frac{16}{25}\).
• This concept is integral when dealing with negative exponents because it transforms the expression to make it easier to evaluate, such as moving to square roots.

Understanding reciprocals is essential especially when working with expressions involving exponents like in our original exercise. Once you've mastered this, tackling more extensive expressions becomes easier.

Square Root

Square roots might bring back classroom memories where you learned about the magic of finding numbers that multiply by themselves to form another number. In mathematics, they are represented by the \(\sqrt{}\) symbol or a \(\frac{1}{2}\) exponent. Here's how it works!

• The square root of a number \(x\) is a number \(y\) such that \(y^2 = x\).
• When given a fractional base, like \(\left(\frac{16}{25}\right)^{1/2}\), you find \(\sqrt{16}\) and \(\sqrt{25}\) separately.
• These become \(4\) and \(5\), respectively, giving you \(\frac{4}{5}\).

Taking the square root simplifies the expression greatly, paving the way to your final simplified answer. It's as if you are rolling back the effect of squaring a number.

Fraction Simplification

Simplifying fractions is all about reducing them to their most basic form, ensuring they are easy to understand and work with. Let's simplify this process:

• Start by identifying if the numerator and the denominator have any common factors.
• If they do, divide both by the greatest common divisor (GCD) to get the simplest form.
• Applying this to the fraction \(\frac{4}{5}\), you will find it is already in its simplest form, as \(4\) and \(5\) share no common factors other than \(1\).

This step makes fractions more manageable and makes the operation results far neater. It comes in handy in algebra and everyday calculations, ensuring clarity and simplicity in mathematical communication.