Problem 45 - Applied Calculus

Question

Evaluate each expression without using a calculator. $$ \left(\frac{25}{16}\right)^{-3 / 2} $$

Step-by-Step Solution

Answer

\( \frac{64}{125} \)

Step 1: Understanding the Expression

We start with the expression \( \left(\frac{25}{16}\right)^{-3/2} \). Our goal is to simplify this without a calculator. We observe that the expression involves a fractional base \( \frac{25}{16} \) raised to a negative fractional exponent.

Step 2: Handling the Negative Exponent

The negative exponent \(-3/2\) indicates that we should take the reciprocal of the base and then apply the exponent. So, \( \left(\frac{25}{16}\right)^{-3/2} = \left(\frac{16}{25}\right)^{3/2} \).

Step 3: Simplifying the Fractional Exponent

The exponent \(\frac{3}{2}\) implies a two-step operation: the denominator \(2\) corresponds to taking the square root, and the numerator \(3\) corresponds to cubing. We will first take the square root of \(\frac{16}{25}\) and then cube the result.

Step 4: Taking the Square Root

Calculate \( \sqrt{\frac{16}{25}} \). This is the same as \( \frac{\sqrt{16}}{\sqrt{25}} = \frac{4}{5} \).

Step 5: Applying the Cube

Now raise \( \frac{4}{5} \) to the power of 3: \( \left(\frac{4}{5}\right)^3 = \frac{4^3}{5^3} = \frac{64}{125} \).

Step 6: Conclusion

The simplified value of the expression \( \left(\frac{25}{16}\right)^{-3/2} \) is \( \frac{64}{125} \).

Key Concepts

Negative Exponents

Understanding negative exponents can make evaluating expressions much simpler. A negative exponent, such as \(-3/2\), tells us that we need to take the reciprocal of the base before we apply the positive exponent. In general, for any non-zero number \((a^{-n} = \frac{1}{a^n})\).
Here's the process in our exercise:

• The base, \(\frac{25}{16}\), has a negative exponent: \(-3/2\).
• This means we take the reciprocal of \(\frac{25}{16}\), which is \(\frac{16}{25}\).
• Now, we apply the exponent \(3/2\) to the reciprocal.

By thinking of negative exponents as a way to "flip" the base, we can easily tackle expressions like these.

Simplifying Expressions

Simplifying expressions requires us to break them down into smaller, more manageable parts. This makes them easier to work with and comprehensible. Our expression \(\left(\frac{25}{16}\right)^{-3/2}\) involves both a fractional base and a fractional exponent, compounding the complexity.
Here are some simplification tips:

• Start by addressing any negative exponents, as we did by flipping the base to \(\frac{16}{25}\).
• Break down the fractional exponent to understand which mathematical operations are involved, such as squaring or cubing.
• Perform each operation in a step-by-step manner, checking your work as you go.

By carefully simplifying expressions in stages and performing small, straightforward calculations, complex expressions become much less intimidating.

Square Roots

Square roots are one of the key steps in working with fractional exponents. The square root of a number is simply a value that, when multiplied by itself, equals the original number. In our exercise, we had to find the square root of a fraction, \(\frac{16}{25}\).
Here's how it's done:

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

Square roots simplify the expression by reducing a power of 2, which can be a great help in calculations, especially when dealing with fractional values.

Rational Exponents

An exponent expressed as a fraction is called a rational exponent. Rational exponents allow us to express roots and powers using a single expression. The exponent \(\frac{3}{2}\) in our problem represents two operations: taking a square root and then cubing the result.
This is how the process unfolds:

• The denominator (2) indicates the root, specifically, the square root.
• The numerator (3) indicates that you should cube the result after finding the root.
• First, find \(\sqrt{\frac{16}{25}} = \frac{4}{5}\).
• Then, raise \(\frac{4}{5}\) to the third power, \(\left(\frac{4}{5}\right)^3 = \frac{64}{125}\).

Rational exponents offer a unified way to handle roots and powers, simplifying our calculations significantly.