Problem 119 - Elementary and Intermediate Algebra

Question

Evaluate: \(\left(\frac{2}{3}\right)^{3}\)

Step-by-Step Solution

Answer

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

Step 1: Identify the Expression

The expression given is \( \left( \frac{2}{3} \right)^{3} \). This means that the fraction \( \frac{2}{3} \) is raised to the power of 3.

Step 2: Break Down the Exponentiation

Exponentiation means multiplying the base by itself a certain number of times equal to the exponent. For \( \left( \frac{2}{3} \right)^{3} \), you will multiply \( \frac{2}{3} \) by itself three times: \( \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} \).

Step 3: Multiply the Numerators

To multiply the fractions, start by multiplying the numerators: \( 2 \times 2 \times 2 = 8 \). This gives us the numerator of the resulting fraction.

Step 4: Multiply the Denominators

Similarly, multiply the denominators: \( 3 \times 3 \times 3 = 27 \). This gives us the denominator of the resulting fraction.

Step 5: Combine and Simplify (if possible)

Put together the results from Steps 3 and 4. The final fraction is \( \frac{8}{27} \). Since 8 and 27 have no common factors other than 1, \( \frac{8}{27} \) is already in its simplest form.

Key Concepts

Fraction Multiplication

When you multiply fractions, you're essentially scaling one fraction by the value of another. This involves dealing with both numerators and denominators separately. Each fraction has a numerator (the top part) and a denominator (the bottom part).

• To multiply two or more fractions, you multiply their numerators together to get the new numerator.
• Next, you multiply all of the denominators together to form the new denominator.

Imagine you have the fractions \( \frac{a}{b} \) and \( \frac{c}{d} \). Their product is calculated by:
\[ \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \]
This rule applies no matter how many fractions you're multiplying together. Just handle all numerators and denominators separately in a straightforward multiplication, and you'll achieve the resulting fraction.

Simplifying Fractions

Once you have multiplied fractions and arrived at a single fraction, the next step is often to simplify it. Simplifying means reducing the fraction to its smallest equivalent form, making it easier to understand and work with.

• Identify the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by their GCD.

Consider the fraction \( \frac{8}{12} \). The GCD of 8 and 12 is 4. Therefore, divide both by 4:
\[ \frac{8}{12} = \frac{8 \div 4}{12 \div 4} = \frac{2}{3} \]
This process does not change the value of the fraction, just its appearance. Having simplified fractions helps when comparing ratios or performing further operations.

Numerators and Denominators

Understanding the components of a fraction—numerators and denominators—is fundamental in working with fractions. Each fraction is composed of two parts:

• The numerator tells how many parts of a whole you have.
• The denominator tells into how many equal parts the whole is divided.

For example, in the fraction \( \frac{3}{5} \):

• The number 3 (numerator) indicates three parts.
• The number 5 (denominator) shows that these parts are out of five equal pieces of the whole.

Understanding this division in fractions aids in visualization and operation. Knowing that both multiplication and simplification of fractions require separate handling of numerators and denominators is key to mastering fraction operations.