Problem 46 - Prealgebra

Question

The problems below review the material on exponents we have covered previously. Expand and simplify. $$\left(-\frac{3}{5}\right)^{3}$$

Step-by-Step Solution

Answer

The expanded and simplified result is \(-\frac{27}{125}\).

Step 1: Understanding Exponentiation

Exponentiation involves raising a base to the power of an exponent, which means multiplying the base by itself as many times as indicated by the exponent. For this problem, we have the expression \( \left(-\frac{3}{5}\right)^3 \), where \(-\frac{3}{5}\) is the base and 3 is the exponent.

Step 2: Write Out the Multiplication

To solve \( \left(-\frac{3}{5}\right)^3 \), we need to multiply \(-\frac{3}{5}\) by itself three times: \(-\frac{3}{5} \times -\frac{3}{5} \times -\frac{3}{5} \).

Step 3: Multiply the First Two Fractions

First, multiply the first two fractions: \(-\frac{3}{5} \times -\frac{3}{5} = \frac{9}{25} \). Since both bases are negative, their product is positive because a negative times a negative equals a positive.

Step 4: Multiply the Result by the Third Fraction

Next, multiply the result \( \frac{9}{25} \) by the remaining fraction: \( \frac{9}{25} \times -\frac{3}{5} \). This equals \(-\frac{27}{125} \), because a positive times a negative equals a negative.

Key Concepts

Exponentiation

In mathematics, exponentiation is a way to express repeated multiplication of a number, known as the base, by itself. To perform exponentiation, you take a base number and raise it to an exponent, indicating how many times the base is multiplied by itself.
For example, in the expression \( \left(-\frac{3}{5}\right)^3 \), the base is \(-\frac{3}{5}\), and the exponent is 3. This means \(-\frac{3}{5}\) needs to be multiplied by itself a total of three times:

• One time gives us \(-\frac{3}{5}\).
• Two times means \(-\frac{3}{5} \times -\frac{3}{5}\).
• Three times involves another multiplication \(\times -\frac{3}{5}\).

Understanding this concept is key to simplifying such expressions effectively.

Fraction Multiplication

Multiplying fractions is a crucial skill when dealing with expressions involving exponentiation of fractional bases. To multiply fractions, follow these basic steps:

• Multiply the numerators (top numbers) together.
• Multiply the denominators (bottom numbers) together.
• Simplify the resulting fraction, if necessary.

In the problem \(\left(-\frac{3}{5}\right)^3\), you first multiply the fractions \(-\frac{3}{5} \times -\frac{3}{5}\), which gives \(\frac{9}{25}\). Then, multiply this result by \(-\frac{3}{5}\) again to yield \(-\frac{27}{125}\).
Remember, multiplying fractions follows the straightforward, step-by-step process, making it manageable even with negative numbers involved.

Negative Bases

When you're dealing with a negative base in exponentiation, it adds an extra layer to the calculation. The result can change depending on whether the exponent is even or odd.

• If the exponent is even, the negative signs will pair up, resulting in a positive product.
• If the exponent is odd, one negative will remain unpaired, making the product negative.

In the example \(\left(-\frac{3}{5}\right)^3\), which features a negative base raised to an odd exponent, the resulting product is negative. This is because the three negative signs do not completely cancel each other out, leaving a single negative sign in the result.
Being aware of how negative bases interact with exponents is crucial for correctly determining the sign of your final answer in such calculations.