Problem 85 - Prealgebra

Question

The problems below review some of the material on solving equations. Reviewing these problems will help you with the next section. Solve. $$\left(\frac{1}{3}\right)^{4}$$

Step-by-Step Solution

Answer

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

Step 1: Understand the problem

The problem asks you to calculate \( \left(\frac{1}{3}\right)^{4} \). This involves applying the rules of exponents to a fraction.

Step 2: Apply exponent rules

Using the rule \( (a/b)^n = a^n / b^n \), raise both the numerator and the denominator to the power of 4. Here, \( a = 1 \) and \( b = 3 \) with \( n = 4 \).

Step 3: Calculate numerator and denominator powers

Raise the numerator 1 to the power of 4: \( 1^4 = 1 \). Then, raise the denominator 3 to the power of 4: \( 3^4 = 81 \).

Step 4: Write the simplified fraction

The result is \( \frac{1}{81} \), since both the numerator and denominator have been raised to the power of 4.

Key Concepts

Understanding Fractions

Fractions are a way to represent numbers that aren't whole. They show us parts of a whole, like splitting a pie into slices. A fraction is made up of two parts:

• The **numerator**: The top number, which represents how many parts you have.
• The **denominator**: The bottom number, which shows the total number of equal parts the whole is divided into.

For example, in the fraction \( \frac{1}{3} \), "1" is the numerator, and "3" is the denominator. It tells us that if you divide something into three equal parts, you have one of those parts. When you see a fraction raised to a power, as in this exercise, you apply the exponent to both the numerator and the denominator. For instance, \( \left(\frac{1}{3}\right)^4 \) means raising both 1 and 3 to the power of 4.

Simplifying Expressions with Exponents

Simplifying expressions is all about making them easier to work with. When you see an expression with exponents, like \( \left(\frac{1}{3}\right)^4 \), you need to apply the exponent rules to simplify it.One important rule is that when you have a fraction elevated to a power, you raise both parts of the fraction to that power. This means if we have \( (a/b)^n \), we know that:

• \( a^n \) is the numerator raised to the power \( n \)
• \( b^n \) is the denominator raised to the power \( n \)

By applying the exponent to each part separately, the fraction stays in the same form, but with each component adjusted by the power. By doing this, we simplify the fraction step by step. For example, \( \left(\frac{1}{3}\right)^4 \), becomes \( \frac{1^4}{3^4} \), simplifying further to \( \frac{1}{81} \). This method makes complex fractions easier to understand and work with.

Solving Equations Involving Fractional Exponents

When solving equations that involve exponents and fractions, it's essential to understand the rules of powers. These rules will help simplify the process, making it easier to find solutions.Equations with fractional exponents usually involve these steps:

• **Identify**: Look at the equation and identify which number is the base and which is the exponent.
• **Apply Rules**: Use the expression rules \((a/b)^n = a^n/b^n\) to simplify. Each part of the fraction is treated separately and raised to the power.
• **Simplify**: This step involves calculating powers for both the numerator and the denominator. Simplifying these numbers individually helps break down the problem into manageable parts.

Using the given problem, \( \left(\frac{1}{3}\right)^4 \), applying the rules gives us \( \frac{1}{81} \). This is the simplified solution of the equation. It shows that understanding and applying these rules precisely leads to accurate answers. The more you practice these steps, the more automatic the process becomes.