Problem 12 - Applied Calculus

Question

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

Step-by-Step Solution

Answer

\( \left(\frac{2}{3}\right)^{-3} = \frac{27}{8} \).

Step 1: Understand the Problem

The problem requires us to evaluate the expression \( \left(\frac{2}{3}\right)^{-3} \) without using a calculator. The expression involves a fraction raised to a negative exponent.

Step 2: Apply the Negative Exponent Rule

Recall that any expression \(a^{-n}\) is equivalent to \( \frac{1}{a^n} \). Therefore, applying this rule gives us \( \left(\frac{2}{3}\right)^{-3} = \frac{1}{\left(\frac{2}{3}\right)^3} \).

Step 3: Simplify the Fraction Raised to an Exponent

Next, evaluate the expression \( \left(\frac{2}{3}\right)^3 \), which means multiplying the fraction by itself two more times: \( \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} \). This simplifies to \( \frac{2^3}{3^3} \).

Step 4: Calculate the Numerator and Denominator

Calculate the values of the numerator and the denominator separately: \( 2^3 = 2 \times 2 \times 2 = 8 \) and \( 3^3 = 3 \times 3 \times 3 = 27 \). Thus, \( \left(\frac{2}{3}\right)^3 = \frac{8}{27} \).

Step 5: Conclude the Evaluation

Finally, from Step 2, we have \( \frac{1}{\left(\frac{2}{3}\right)^3} = \frac{1}{\frac{8}{27}} = \frac{27}{8} \). Therefore, \( \left(\frac{2}{3}\right)^{-3} = \frac{27}{8} \).

Key Concepts

Fraction Exponentiation

Fraction exponentiation involves raising a fraction to a given power. In fraction exponentiation, we deal with both the numerator and the denominator separately. Opposed to regular numbers, which you multiply by themselves a certain number of times, things are more interesting with fractions.
When you see an expression like \( \left(\frac{2}{3}\right)^3 \), the idea is to raise both the numerator \(2\) and the denominator \(3\) to the power of \(3\). This means you multiply \(2\) by itself three times and \(3\) by itself three times.

• This gives us \(2^3 = 8\) for the numerator.
• Similarly, \(3^3 = 27\) for the denominator.

So, \( \left(\frac{2}{3}\right)^3 = \frac{8}{27} \). By knowing this, you can handle fractions just like regular numbers when using exponents.

Exponent Rules

Exponent rules are crucial when working with powers and exponents. They help us simplify expressions that look complex at first glance. One of the key rules to understanding in this exercise is the negative exponent rule.
The negative exponent rule states that a number raised to a negative exponent is equal to the reciprocal of the number raised to the positive exponent. This is expressed as \( a^{-n} = \frac{1}{a^n} \).
When you encounter negative exponents, flip the base of the fraction. Therefore:

• \( \left(\frac{2}{3}\right)^{-3} \) becomes \( \frac{1}{\left(\frac{2}{3}\right)^3} \)
• It means you are effectively inverting the fraction and raising it to a positive exponent.

This rule allows one to transform negative exponents into fractions, paving the way for clearer and simpler calculations.

Simplification of Fractions

Simplifying fractions is the process of reducing fractions to their simplest form. This concept is especially important when dealing with complex algebraic expressions and negative exponents.
To simplify a fraction, you divide both the numerator and the denominator by their greatest common divisor (GCD). However, in many problems like this one, simplification involves working backwards from a reciprocal.
For example, once you find that \( \left(\frac{2}{3}\right)^3 = \frac{8}{27} \), the next task is to find the reciprocal, giving you \( \left(\frac{2}{3}\right)^{-3} = \frac{27}{8} \).

• This ensures the fraction cannot be simplified further.
• It provides a check that the operation of inverting worked correctly.

Simplification helps to present any expression in its most efficient form, aiding in immediate understanding and further calculations.