Problem 80 - Prealgebra

Question

Use the definition of exponents to simplify each expression. \(\left(\frac{1}{3}\right)^{4}\)

Step-by-Step Solution

Answer

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

Step 1: Understanding the Base and Exponent

Recognize that the expression contains a fraction as the base, which is \( \frac{1}{3} \), raised to an exponent of 4. This means you will multiply the base by itself four times.

Step 2: Multiply the Fraction by Itself

Multiply \( \frac{1}{3} \times \frac{1}{3} \). This gives \( \frac{1 \times 1}{3 \times 3} = \frac{1}{9} \).

Step 3: Repeat the Multiplication of Fractions

Now, take the result from the previous step \( \frac{1}{9} \) and multiply by \( \frac{1}{3} \) again: \( \frac{1}{9} \times \frac{1}{3} = \frac{1 \times 1}{9 \times 3} = \frac{1}{27} \).

Step 4: Final Multiplication

Finally, multiply the result \( \frac{1}{27} \) by \( \frac{1}{3} \) one more time: \( \frac{1}{27} \times \frac{1}{3} = \frac{1 \times 1}{27 \times 3} = \frac{1}{81} \).

Step 5: Conclusion

The expression \( \left(\frac{1}{3}\right)^4 \) simplifies to \( \frac{1}{81} \).

Key Concepts

Exponentiation of Fractions

The process of exponentiating a fraction may seem challenging at first, but it becomes straightforward with correct understanding and practice. When you have a fraction like \( \left(\frac{1}{3}\right)^4 \), the exponent indicates how many times you multiply the base by itself. In this case, \( \frac{1}{3} \) is your base. With an exponent of 4, you will multiply this fraction by itself four times.

• The result of the exponentiation tells us how many times to repeatedly multiply the base.
• Exponentiation of fractions follows the same principles as whole numbers.
• Simply multiply the fraction by itself as many times as the exponent indicates.

By exponentiating the fraction \( \frac{1}{3} \) four times, it ultimately simplifies to \( \frac{1}{81} \). Every multiplication simplifies the fraction further until you reach the final simplified result.

Multiplication of Fractions

Multiplying fractions is an essential skill needed for simplifying expressions like \( \left(\frac{1}{3}\right)^4 \). When multiplying fractions, you multiply the numerators together and the denominators together. This process applies each time you multiply to effectively handle fraction multiplication.

• To start, take the fraction \( \frac{1}{3} \) and multiply the numerators: \( 1 \times 1 = 1 \).
• Next, multiply the denominators: \( 3 \times 3 = 9 \).
• The result from the first multiplication gives you \( \frac{1}{9} \).

Continuing this method, multiply \( \frac{1}{9} \) by \( \frac{1}{3} \) to get \( \frac{1}{27} \), then \( \frac{1}{27} \) again by \( \frac{1}{3} \) for the final answer, \( \frac{1}{81} \). Always remember this simple technique of multiplying across to ensure accuracy in reducing fractions.

Base and Exponent Concept

Grasping the base and exponent concept is critical for simplifying expressions with exponents, particularly when dealing with fractions. The base is the number or fraction you multiply, and the exponent tells you how many times to perform this multiplication. In our case, \( \frac{1}{3} \) is the base, and 4 is the exponent.

• The base is what is being multiplied repeatedly.
• The exponent signals the number of times the base is used as a factor in the multiplication.
• Understanding these two parts aids in visualizing and executing the process of exponentiation.

For an expression like \( \left(\frac{1}{3}\right)^4 \), you take the base \( \frac{1}{3} \) and multiply it by itself four times as directed by the exponent. Each multiplication step simplifies the expression, leading to the final result. This clear breakdown of base and exponent concepts is crucial in systematically solving exponent problems.