Problem 86 - Prealgebra

Question

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

Step-by-Step Solution

Answer

The value of \( \left(\frac{3}{4}\right)^3 \) is \( \frac{27}{64} \).

Step 1: Understanding Exponents

The expression \( \left(\frac{3}{4}\right)^3 \) means that the fraction \( \frac{3}{4} \) is multiplied by itself three times.

Step 2: Write the Multiplication

Since \( \left(\frac{3}{4}\right)^3 \) is a power of three, write it as \( \frac{3}{4} \times \frac{3}{4} \times \frac{3}{4} \).

Step 3: Multiply the Fractions - Numerator

Multiply the numerators: \( 3 \times 3 \times 3 = 27 \). This will be the numerator of the resulting fraction.

Step 4: Multiply the Fractions - Denominator

Multiply the denominators: \( 4 \times 4 \times 4 = 64 \). This will be the denominator of the resulting fraction.

Step 5: Write the Final Answer

Combine the results from Steps 3 and 4 to get the final fraction: \( \frac{27}{64} \).

Key Concepts

Exponents

When exploring the realm of exponents, you're essentially dealing with repeated multiplication of the same number. An exponent is a small, raised number next to another number that tells you how many times to use that number in a multiplication. For example, in our exercise, the expression \( \left(\frac{3}{4}\right)^3 \) indicates that the fraction \( \frac{3}{4} \) should be multiplied by itself three times.

Understanding exponents is crucial when solving equations, as it simplifies expressions and allows you to assess repeated operations with ease. Instead of writing \( \frac{3}{4} \times \frac{3}{4} \times \frac{3}{4} \), the exponent neatly condenses this action into a single, power-packed notation.

Multiplying Fractions

Multiplying fractions might seem complex at first, but once you break down the process, it's quite straightforward. The beauty of fractions is in their simplicity when it comes to multiplication. Unlike addition or subtraction, you do not need a common denominator here! Instead, you multiply the numerators together to get the new numerator, and then multiply the denominators together to get the new denominator.

For the problem \( \frac{3}{4} \times \frac{3}{4} \times \frac{3}{4} \), we multiply the numerators: \( 3 \times 3 \times 3 = 27 \). Next, we multiply the denominators: \( 4 \times 4 \times 4 = 64 \). This means our final multiplication results in the fraction \( \frac{27}{64} \), beautifully combining both actions into a single operation.

Numerator and Denominator Multiplication

Understanding how to multiply numerators and denominators separately is key when dealing with fraction multiplication. Here’s how you can see it in a clearer light:

• **Numerator Multiplication:** This is simply multiplying the numbers on the top of each fraction. For example, when you take \( 3 \times 3 \times 3 \), you just multiply these directly to get 27.
• **Denominator Multiplication:** Similarly, this involves multiplying the numbers on the bottom of each fraction. Here, substituting \( 4 \times 4 \times 4 \) results in 64.

By segregating these stages, it becomes easier to grasp the flow of multiplying fractions. Each set of numbers remains in its domain, ensuring you seamlessly find the resulting fraction from their multiplication. This separation of tasks is what ultimately leads to the simplified fraction result, \( \frac{27}{64} \), which is a representation of multiplied power.