Problem 11 - Algebra A Combined Function

Question

Evaluate. $$ \left(\frac{2}{3}\right)^{4} $$

Step-by-Step Solution

Answer

\( \left(\frac{2}{3}\right)^{4} = \frac{16}{81} \)

Step 1: Understand the Expression

The expression \( \left(\frac{2}{3}\right)^{4} \) is asking for you to raise the fraction \( \frac{2}{3} \) to the power of 4. This means we need to multiply \( \frac{2}{3} \) by itself four times.

Step 2: Multiply Fractions

When multiplying fractions, multiply the numerators together and the denominators together. Our problem is: \[ \left(\frac{2}{3}\right)^{4} = \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} \]

Step 3: Multiply the Numerators

First, multiply all the numerators of the fractions: \[ 2 \times 2 \times 2 \times 2 = 16 \]

Step 4: Multiply the Denominators

Next, multiply all the denominators of the fractions: \[ 3 \times 3 \times 3 \times 3 = 81 \]

Step 5: Simplify the Fraction

Now take the results from Steps 3 and 4 and put them together to form the fraction: \[ \frac{16}{81} \] This fraction is already in its simplest form since 16 and 81 have no common factors.

Key Concepts

Understanding Fractions

Fractions represent parts of a whole and are written in the form \(\frac{a}{b}\). Here, \(a\) is the numerator, and \(b\) is the denominator—both are integers. Fractions express values less than one, where the numerator is smaller than the denominator, or greater than or equal to one if the numerator is larger or equal to the denominator.

Some ways fractions are used include dividing pizza amongst friends or describing any portioned quantity. Understanding how to operate with fractions involves both basic arithmetic and an understanding that the numerator depicts how many parts you have while the denominator depicts how many parts make up a whole.

• The larger the denominator, the smaller each part is.
• If the numerator equals the denominator, the fraction is equivalent to 1.

Fractions can also extend into more complex operations such as being raised to a power, like in our exercise \(\left(\frac{2}{3}\right)^{4}\). By grasping the basic concept of fractions, tackling exercises becomes much more intuitive.

Multiplying Fractions

Multiplying fractions involves a straightforward operation, where the numerators and denominators are multiplied together. Let's break it down using the expression \(\left(\frac{2}{3}\right)^{4}\):

• First, write the fraction being multiplied by itself as many times as indicated by the exponent. For instance, here it means \(\frac{2}{3} \times \frac{2}{3} \times \frac{2}{3} \times \frac{2}{3}\).
• Multiply the numerators together: \(2 \times 2 \times 2 \times 2 = 16\).
• Multiply the denominators: \(3 \times 3 \times 3 \times 3 = 81\).

The result of multiplying fractions is simply placing the product of the numerators over the product of the denominators, which gives us \(\frac{16}{81}\). Through clear procedures and practice, multiplying fractions becomes an easier task. Each multiplication step must adhere to the rules of keeping operations across the numerators and denominators separate at first.

Simplifying Fractions

Simplifying fractions is the process of reducing them to their simplest form, where the numerator and denominator have no common divisors other than 1. A fraction is in simplest form when its numerator and denominator are coprime.

In our exercise, after multiplying, we arrive at \(\frac{16}{81}\). To check if a fraction is simplified, find the greatest common factor (GCF) of the numerator and the denominator:

• List the factors: For 16, the factors are 1, 2, 4, 8, 16; for 81, they are 1, 3, 9, 27, 81.
• Identify common factors: Here, the only common factor is 1.

Since the GCF is 1, \(\frac{16}{81}\) is already in its simplest form. Simplifying helps in making fractions more understandable and easy to work with, especially when solving math problems or comparing fractions.