Problem 202 - Intermediate Algebra

Question

In the following exercises, simplify. $$ \frac{3^{3}-3^{2}}{\left(\frac{3}{4}\right)^{2}} $$

Step-by-Step Solution

Answer

32

Step 1: Evaluate the numerator

First, calculate the expressions in the numerator. Simplify both terms individually: \( 3^3 = 27 \) and \( 3^2 = 9 \). Then subtract these values: \[ 27 - 9 = 18 \]

Step 2: Evaluate the denominator

Next, simplify the expression in the denominator: \[ \frac{3}{4} \] Raise this fraction to the power of 2:\( \left( \frac{3}{4} \right)^2 \), which gives:\[ \frac{9}{16} \]

Step 3: Divide the numerator by the denominator

Now, divide the simplified numerator by the simplified denominator:\[ \frac{18}{\left( \frac{9}{16}\right)} \]This is equivalent to:\[ 18 \times \frac{16}{9} \]Simplify the multiplication:\[ 18 \times \frac{16}{9} = 2 \times 16 = 32 \]

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number to the power of another number. In simple terms, if you see something like \(a^n\), it means you multiply \(a\) by itself \(n\) times.
For example, in the problem:

• \(3^3\)
• This means you multiply 3 by itself 3 times: \(3 \times 3 \times 3 = 27\).

Another example from the exercise is \(3^2\). Here, we multiply 3 by itself 2 times: \(3 \times 3 = 9\). By understanding these basics, you can evaluate any exponential expression. The general formula is:

• If \(a\) is any number and \(n\) is a positive integer, then \(a^n\) means you multiply \(a\) by itself \(n\) times.

Fractions

Fractions represent parts of a whole and have two main parts: the numerator and the denominator. In mathematical expressions, fractions can often appear within exponentiation or division problems.
From the exercise, we see the fraction \(\left( \frac{3}{4} \right)^2\). This fraction means we are raising \(\frac{3}{4}\) to the power of 2, multiplying the fraction by itself:

• \(\left( \frac{3}{4} \right)^2 = \frac{3}{4} \times \frac{3}{4} = \frac{9}{16}\)

When you have fractions inside an exponent, you simply apply the exponent to both the numerator and the denominator.

Numerator and Denominator

Understanding the parts of a fraction is core to simplifying it correctly. The numerator is the top number and represents how many parts you have. The denominator is the bottom number and represents into how many parts the whole is divided.
In the given problem, once we simplified the numerator and the denominator:

• The numerator became 18
• The denominator became \(\frac{9}{16}\)

To divide the numerator by the denominator: \(\frac{18}{\frac{9}{16}}\), change this to a multiplication problem by flipping the fraction in the denominator:
\(18 \times \frac{16}{9}\). This results in:

• \(18 \times \frac{16}{9} = 2 \times 16 = 32\)

By understanding how to work with numerators and denominators, you can easily simplify complex fractions.