Problem 28 - Brief Applied Calculus

Question

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

Step-by-Step Solution

Answer

The expression evaluates to \( \frac{64}{125} \).

Step 1: Understand the Expression

The expression \( \left(\frac{16}{25}\right)^{3/2} \) involves a fractional BASE raised to a fractional EXPONENT. Let's break this down step by step.

Step 2: Simplify the Base

First, recognize that 16 and 25 are perfect squares. We can express them as \( 16 = 4^2 \) and \( 25 = 5^2 \). Thus, \( \frac{16}{25} = \left(\frac{4}{5}\right)^2 \).

Step 3: Apply the Power of a Power Rule

When you raise a power to another power, you multiply the exponents. The expression becomes: \[ \left(\frac{4}{5}\right)^{2 \times \frac{3}{2}} \] which simplifies to \[ \left(\frac{4}{5}\right)^{3} \].

Step 4: Simplify the Cubed Fraction

Now, raise both the numerator and the denominator of the fraction to the power of 3. This gives us: \[ \frac{4^3}{5^3} = \frac{64}{125} \].

Key Concepts

Perfect Squares

Perfect squares are numbers that can be expressed as the square of an integer. In simpler terms, a perfect square is a number that results from multiplying an integer by itself. For example, 16 is a perfect square because it's equivalent to

• \( 4 \times 4 = 16 \)

and 25 is a perfect square because

• \( 5 \times 5 = 25 \).

In the original exercise, it was crucial to recognize that both 16 and 25 are perfect squares, enabling us to express them as powers of 2. This understanding simplifies our problem by allowing the base of the fractional expression \( \left(\frac{16}{25}\right) \) to be written as \( \left(\frac{4}{5}\right)^2 \), making it easier to apply subsequent mathematical rules.

Power of a Power Rule

The power of a power rule is a helpful exponent rule for simplifying expressions. When a base with an exponent is raised to another exponent, you can multiply the two exponents together. This rule is very useful, especially in expressions that involve more than one layer of exponents.

For example, in our expression \( \left(\frac{4}{5}\right)^{2 \times \frac{3}{2}} \), we are raising \( \left(\frac{4}{5}\right)^2 \) to the power of \( \frac{3}{2} \). According to the power of a power rule, you multiply the exponents:

• \( 2 \times \frac{3}{2} = 3 \).

This results in \( \left(\frac{4}{5}\right)^3 \), simplifying the calculation. Understanding and applying this rule will make dealing with complex expressions much easier.

Cubed Fraction

A cubed fraction involves raising both the numerator and the denominator of a fraction to the power of three. When you encounter a cubed fraction, you simply cube each part separately to get the result.

In our exercise, once we simplified \( \left(\frac{4}{5}\right)^3 \), we need to cube both the numerator and the denominator:

• The numerator is cubed as \( 4^3 = 64 \).
• The denominator is cubed as \( 5^3 = 125 \).

After performing these calculations, the expression simplifies to \( \frac{64}{125} \). Cubing fractions may seem tricky at first, but by breaking it down, it becomes quite straightforward.