Problem 48 - Prealgebra

Question

Expand and simplify each of the following. $$\left(-\frac{3}{5}\right)^{3}$$

Step-by-Step Solution

Answer

\((-\frac{3}{5})^{3} = \frac{-27}{125}\).

Step 1: Understanding the Problem

The given expression is \(\left(-\frac{3}{5}\right)^{3}\). This asks us to raise \(-\frac{3}{5}\) to the power of 3. This operation involves multiplying \(-\frac{3}{5}\) by itself three times.

Step 2: Breaking Down the Multiplication

We can write \(\left(-\frac{3}{5}\right)^{3}\) as \((-\frac{3}{5}) \times (-\frac{3}{5}) \times (-\frac{3}{5})\). Now it's easier to handle, since we will multiply the fractions step-by-step.

Step 3: First Multiplication

Multiply the first two fractions: \((-\frac{3}{5}) \times (-\frac{3}{5})\). Multiply the numerators and the denominators separately: numerator \(-3 \times -3 = 9\), denominator \(5 \times 5 = 25\). So, \((-\frac{3}{5}) \times (-\frac{3}{5}) = \frac{9}{25}\).

Step 4: Second Multiplication

Now, multiply this result by \(-\frac{3}{5}\): \(\frac{9}{25} \times (-\frac{3}{5})\). Again, multiply numerators: \(9 \times -3 = -27\), and denominators: \(25 \times 5 = 125\). Thus, \(\frac{9}{25} \times (-\frac{3}{5}) = \frac{-27}{125}\).

Step 5: Interpretation of the Result

Finally, simplify the expression. Since \(-\frac{27}{125}\) is already in its simplest form, this is our final answer.

Key Concepts

Fraction Multiplication

When multiplying fractions, the process involves multiplying the numerators together and the denominators together.
Imagine each fraction is a small part of a larger whole. Combining them, by multiplication, combines their components together.Let's apply this to the example we have:
We start with two fractions: \(-\frac{3}{5}\) multiplied by \(-\frac{3}{5}\).
You multiply the top numbers (numerators) \(-3 \times -3\), which equals \(9\).
Then, multiply the bottom numbers (denominators) \(5 \times 5\), which equals \(25\). So, the product is \(\frac{9}{25}\).
Next, you multiply the result \(\frac{9}{25}\) by the remaining fraction \(-\frac{3}{5}\). Again, multiply the numerators and denominators:

• Numerators: \(9 \times -3 = -27\)
• Denominators: \(25 \times 5 = 125\)

Therefore, the final result is \(\frac{-27}{125}\).
Dividing fractions can be intimidating at first, but breaking it down step-by-step helps simplify the process.

Negative Numbers

Negative numbers can change the outcome of our calculations significantly.
You can think of them like arrows pointing in the opposite direction:Multiplying two negative numbers results in a positive number.
When you multiply a negative by a positive or vice versa, the result is a negative.
For our expression \((-\frac{3}{5})^3\):

• The first multiplication of two negatives: \( (-3) \times (-3) = 9 \)
• The second multiplication introduces a negative once more: \( 9 \times (-3) = -27 \)

Notice that each time we involve an odd number of negative signs, the overall product remains negative.
Understanding how negative signs function within multiplication helps us keep track of the final sign of our result.

Simplifying Fractions

Simplifying fractions is the final touch to rounding off your solution.
It's all about reducing the fraction to its simplest form, where the numerator and denominator have no common divisors except for 1.In our example, we ended with the fraction \(\frac{-27}{125}\):
Finding the largest number both the numerator (-27) and the denominator (125) can be divided by helps simplify:

• The number 27 has factors: 1, 3, 9, 27
• The number 125 has factors: 1, 5, 25, 125

The greatest common factor is 1.
Therefore, the fraction \(\frac{-27}{125}\) is already in its simplest form. Simplification ensures that every fraction is presented in its simplest, clearest way, making it easier to compare and work with.
Always strive to reduce fractions where possible to ensure clear and concise results.