Problem 73 - Algebra A Combined Function

Question

Simplify each expression. $$ \left(\frac{1}{4}\right)^{3} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{1}{64} \).

Step 1: Understand the Expression

The given expression is \( \left(\frac{1}{4}\right)^3 \). This is a power expression where \( \frac{1}{4} \) is the base and 3 is the exponent. The expression \( \left(\frac{1}{4}\right)^3 \) means \( \frac{1}{4} \times \frac{1}{4} \times \frac{1}{4} \).

Step 2: Multiply the Fractions

To simplify \( \left(\frac{1}{4}\right)^3 \), multiply the base \( \frac{1}{4} \) by itself three times: \( \frac{1}{4} \times \frac{1}{4} = \frac{1}{16} \), and then multiply by \( \frac{1}{4} \) again: \( \frac{1}{16} \times \frac{1}{4} = \frac{1}{64} \).

Step 3: Final Simplified Expression

The simplified expression of \( \left(\frac{1}{4}\right)^3 \) is \( \frac{1}{64} \).

Key Concepts

Understanding Fractions

Fractions are a way of representing numbers that are not whole. They consist of two parts: a numerator and a denominator. The numerator is the number on top, indicating how many parts we have. The denominator is the number at the bottom, showing how many parts make up a whole. For example, in the fraction \( \frac{1}{4} \), 1 is the numerator and 4 is the denominator. It represents one part out of four equal parts of a whole.
Working with fractions can sometimes be tricky, especially when it involves operations like multiplication or exponentiation. But understanding the basic structure of fractions will help simplify these processes.

• The numerator (top part) indicates how many parts we are considering.
• The denominator (bottom part) tells us how many parts make up a whole

Multiplying Fractions

Multiplying fractions is an important math skill, and it's simpler than it might initially seem. To multiply two fractions, multiply the numerators together to get a new numerator, and the denominators together to get a new denominator. For example, \( \frac{1}{4} \times \frac{1}{4} \) becomes \( \frac{1 \times 1}{4 \times 4} = \frac{1}{16} \).
When you have more than two fractions, like in the problem \( \left(\frac{1}{4}\right)^3 = \frac{1}{4} \times \frac{1}{4} \times \frac{1}{4} \), you simply continue the process. Multiply the result of the first multiplication (\( \frac{1}{16} \)) with the next fraction (\( \frac{1}{4} \)):

• \( \frac{1}{16} \times \frac{1}{4} \) results in \( \frac{1 \times 1}{16 \times 4} = \frac{1}{64} \)

The key to multiplying fractions is to simplify whenever possible to keep the calculations manageable.

Simplification Steps

Simplification is the process of reducing a mathematical expression to its simplest form. This often involves performing multiplication or division and reducing fractions.
After multiplying the fractions, you should check if the resulting fraction can be reduced. A fraction is in its simplest form when the numerator and the denominator have no common factors other than 1. In the expression \( \frac{1}{64} \), the fraction is already simplified because 1 and 64 have no common factors.

• Ensure multiplication steps are correct.
• Check if you can divide both the numerator and denominator by the same number, other than 1, to simplify.

Understanding when to stop simplifying is essential, as sometimes the fraction is already in its simplest form when you finish multiplying.