Problem 29 - Applied Calculus

Question

Evaluate each expression without using a calculator. $$ \left(\frac{27}{125}\right)^{2 / 3} $$

Step-by-Step Solution

Answer

\( \frac{9}{25} \)

Step 1: Understand the Problem

We need to evaluate \( \left(\frac{27}{125}\right)^{2/3} \). This expression involves a fractional base raised to a fractional exponent.

Step 2: Break Down the Exponent

The exponent \( \frac{2}{3} \) means we need to first take the cubic root of the base \( \frac{27}{125} \) and then square the result.

Step 3: Calculate the Cubic Root of the Numerator and Denominator

Find the cubic root of the numerator and the denominator separately. \( \sqrt[3]{27} = 3 \) and \( \sqrt[3]{125} = 5 \), so \( \sqrt[3]{\frac{27}{125}} = \frac{3}{5} \).

Step 4: Square the Result

Now we take the result from the previous step \( \frac{3}{5} \) and square it. \( \left(\frac{3}{5}\right)^2 = \frac{3^2}{5^2} = \frac{9}{25} \).

Step 5: Write the Final Result

The value of \( \left(\frac{27}{125}\right)^{2/3} \) is \( \frac{9}{25} \).

Key Concepts

Numerator and Denominator

In the world of fractions, the numerator and denominator are like a team, working together to form a complete fraction. Let’s dive deep into what each part means.

• Numerator: This is the top part of a fraction. It represents the number of parts we are focusing on or counting.
• Denominator: This is the bottom part of a fraction. It indicates the total number of equal parts the whole is divided into.

Consider the fraction \( \frac{27}{125} \) from our exercise. The numerator is 27, and the denominator is 125. When raising fractions to powers, both parts are treated independently. For example, finding the cubic root or squaring involves dealing with each separately. This approach simplifies calculations, as we handle smaller numbers directly.

Cubic Root

A cubic root is a special type of root in mathematics. It answers the question: "What number can you multiply by itself twice to get this original number?" Taking the cubic root is an important step when dealing with fractional exponents, such as \( \frac{2}{3} \).For instance, in the problem \( \left(\frac{27}{125}\right)^{2/3} \), you first need to extract the cubic root of both the numerator and the denominator:

• The cubic root of 27 is 3, because \( 3 \times 3 \times 3 = 27 \).
• The cubic root of 125 is 5, since \( 5 \times 5 \times 5 = 125 \).

Once you have these simpler values, \( \sqrt[3]{\frac{27}{125}} \) becomes \( \frac{3}{5} \). Now, further operations such as squaring become much easier.

Rational Numbers

Rational numbers are numbers that can be expressed as fractions, where both the numerator and the denominator are integers. A key characteristic is that the denominator cannot be zero. Rational numbers include fractions like \( \frac{27}{125} \) and whole numbers (e.g., 3, which can be written as \( \frac{3}{1} \)).In evaluating expressions with rational numbers, such as \( \left(\frac{27}{125}\right)^{2/3} \), understanding their properties helps. They make it feasible to work through complex expressions since operations like root extraction and exponentiation apply to both numerators and denominators individually.
When working with fractional exponents, these steps reveal the powerful simplicity of rational numbers: they remain rational, as long as operations are appropriately applied to both parts of the fraction. Returning to our example, even after manipulations, the result \( \frac{9}{25} \) stays within the realm of rational numbers.