Problem 88 - Prealgebra

Question

The problems below review some of the material on solving equations. Reviewing these problems will help you with the next section. Solve. $$\left(\frac{3}{5}\right)^{3}$$

Step-by-Step Solution

Answer

The value of \( \left(\frac{3}{5}\right)^{3} \) is \( \frac{27}{125} \).

Step 1: Identify the base and the exponent

The given expression is \( \left(\frac{3}{5}\right)^{3} \). Here, the base is \( \frac{3}{5} \) and the exponent is \( 3 \). This means that we need to multiply \( \frac{3}{5} \) by itself three times.

Step 2: Multiply the base by itself according to the exponent

To evaluate \( \left(\frac{3}{5}\right)^{3} \), we multiply \( \frac{3}{5} \times \frac{3}{5} \times \frac{3}{5} \).

Step 3: Multiply the fractions step-by-step

When multiplying fractions, multiply the numerators together and the denominators together. First, multiply two fractions: \( \frac{3}{5} \times \frac{3}{5} = \frac{3\times3}{5\times5} = \frac{9}{25} \).

Step 4: Complete the multiplication

Now, multiply the result from Step 3 by \( \frac{3}{5} \): \( \frac{9}{25} \times \frac{3}{5} = \frac{9\times3}{25\times5} = \frac{27}{125} \).

Step 5: Simplify the final result

The fraction \( \frac{27}{125} \) is already in its simplest form, as there are no common factors between the numerator and the denominator.

Key Concepts

Understanding Fractions

A fraction represents a part of a whole. It's written as one number over another, separated by a line, such as \( \frac{3}{5} \). The number on top is called the numerator, and it tells you how many parts you have. The number at the bottom is called the denominator, and it tells you into how many equal parts the whole is divided.

Fractions can be thought of as division problems. For example, \( \frac{3}{5} \) can be interpreted as 3 divided by 5. This makes fractions very useful in various mathematical operations, like addition, subtraction, multiplication, and division.

• A fraction with the same numerator and denominator is equal to 1, such as \( \frac{5}{5} = 1 \).
• Improper fractions have numerators larger than denominators, such as \( \frac{7}{4} \).
• Mixed numbers are a combination of a whole number and a fraction.

Recognizing these properties and knowing how to manipulate fractions is key to solving problems that involve fractions, such as those with equations.

Exploring Exponents

Exponents are a way to express repeated multiplication. They are written as a small number placed to the upper right of a base number. If we have \( \left(\frac{3}{5}\right)^{3} \), then \( \frac{3}{5} \) is the base and 3 is the exponent.

The exponent dictates how many times the base is used as a factor in a multiplication. In our case:

• The expression \( \left(\frac{3}{5}\right)^{3} \) means \( \frac{3}{5} \times \frac{3}{5} \times \frac{3}{5} \).
• An exponent of 1 means the base remains unchanged, and an exponent of 0 means any nonzero base raised to this power is 1.

Understanding and working with exponents simplifies equations significantly. The power of exponents is that they provide a compact way to deal with large numbers or situations involving repeated multiplication.

Multiplying Fractions Made Easy

Multiplying fractions is a straightforward process compared to adding or subtracting them. You don't have to worry about finding a common denominator. To multiply fractions like \( \frac{3}{5} \times \frac{3}{5} \), simply multiply the numerators together to get the new numerator, and multiply the denominators together to get the new denominator.

Let's break it down:

• For \( \frac{3}{5} \times \frac{3}{5} \), multiply the numerators: \( 3 \times 3 = 9 \).
• Multiply the denominators: \( 5 \times 5 = 25 \).
• Therefore, \( \frac{3}{5} \times \frac{3}{5} = \frac{9}{25} \).

Repeating the process with another multiplication, \( \frac{9}{25} \times \frac{3}{5} \):

• Numerators: \( 9 \times 3 = 27 \).
• Denominators: \( 25 \times 5 = 125 \).
• The result is \( \frac{27}{125} \).

When multiplying fractions, focus on multiplying straight across, simplifying only if necessary after the final multiplication step. This method saves time and helps avoid errors.