Problem 43 - Prealgebra

Question

The problems below review the material on exponents we have covered previously. Expand and simplify. $$\left(\frac{1}{3}\right)^{4}$$

Step-by-Step Solution

Answer

\(\frac{1}{81}\)

Step 1: Understanding the expression

We are given the expression \(\left(\frac{1}{3}\right)^{4}\). This means that the fraction \(\frac{1}{3}\) is raised to the power of 4, indicating that we need to multiply \(\frac{1}{3}\) by itself 4 times.

Step 2: Multiply the base by itself

Calculate the expression by multiplying \(\frac{1}{3}\) four times: \(\frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3}\).

Step 3: Multiply the fractions

To multiply the fractions, multiply the numerators together and the denominators together. The numerators are all 1, and the denominators are all 3.

Step 4: Calculate the final result

The numerator becomes \(1 \times 1 \times 1 \times 1 = 1\) and the denominator becomes \(3 \times 3 \times 3 \times 3 = 81\). Therefore, the result of \(\left(\frac{1}{3}\right)^{4}\) is \(\frac{1}{81}\).

Key Concepts

Simplifying Expressions

Simplifying expressions means reducing them to a simpler form without changing their value. In the context of exponents, especially fraction exponents, this often involves performing operations like multiplying the base number by itself as many times as indicated by the exponent. Let's consider the expression \( \left(\frac{1}{3}\right)^{4} \). Here, the goal is to express it in its simplest form. To simplify it, we carry out the multiplication \( \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} \). Simplifying often requires the following steps:

• Identify the base and the exponent.
• Multiply the base by itself repeatedly, as many times as indicated by the exponent.
• Combine the results by multiplying the numerators and denominators separately.

Breaking down expressions into these steps makes them easier to handle and understand. Simplifying helps in better understanding the overall value of a complex expression.

Fraction Exponents

Fraction exponents are exponents where the base is a fraction. Fractions present a unique scenario, as they involve both a numerator and a denominator, which each need separate attention when calculating with exponents. Consider \( \left(\frac{1}{3}\right)^{4} \) as the base. In this case, the fraction \( \frac{1}{3} \) is raised to the power of 4. This implies repeating the base fraction four times in a multiplication operation. Handling fraction exponents can be straightforward by following these steps:

• Multiply the fraction by itself as many times as the exponent indicates.
• First, multiply all numerators together.
• Second, multiply all denominators together.

By mastering fraction exponents, you can approach problems involving them with confidence, ensuring accuracy in calculations.

Multiplying Fractions

Multiplying fractions involves multiplying numerators with numerators and denominators with denominators. This process is fundamental in dealing with fraction exponents and simplifies the expression into a single fraction. Let's illustrate with \( \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} \times \frac{1}{3} \). To multiply:

• Multiply the numbers in the numerator: \(1 \times 1 \times 1 \times 1 = 1\).
• Multiply the numbers in the denominator: \(3 \times 3 \times 3 \times 3 = 81\).

Thus, the result is \( \frac{1}{81} \).Remember these key points when multiplying fractions:

• Always simplify before multiplying if possible.
• Ensure that you multiply straight across, not cross-multiply.
• The final fraction should be in its simplest form, but if you follow the steps precisely, it often ends up that way.

Mastering the multiplication of fractions aids in comprehending more complex algebraic expressions and problem-solving scenarios.