Problem 53 - Intermediate Algebra

Question

Evaluate each expression. See Example \(6 .\) $$ \left(-\frac{3}{5}\right)^{3} $$

Step-by-Step Solution

Answer

The evaluated expression is \( -\frac{27}{125} \).

Step 1: Understanding the Expression

The expression given is \( \left(-\frac{3}{5}\right)^{3} \). It involves raising a fraction, which is negative, to the power of 3. This requires us to multiply \( -\frac{3}{5} \) by itself three times.

Step 2: Multiply the Fraction by Itself

Start by multiplying \( -\frac{3}{5} \) by itself: \[ (-\frac{3}{5}) \times (-\frac{3}{5}) = \frac{9}{25} \] Since both terms are negative, their product is positive.

Step 3: Multiply the Result by the Fraction Again

Now, take the result from Step 2, \( \frac{9}{25} \), and multiply it by \( -\frac{3}{5} \) again: \[ \frac{9}{25} \times (-\frac{3}{5}) = -\frac{27}{125} \] Here, the product is negative because a positive number is multiplied by a negative number.

Step 4: Final Evaluation

The final evaluated result of \( \left(-\frac{3}{5}\right)^{3} \) is \( -\frac{27}{125} \). This is because we multiplied \( -\frac{3}{5} \) three times in total.

Key Concepts

Negative fractions

Understanding negative fractions is important when dealing with exponentiation, especially when it comes to raising them to a power. A negative fraction is simply a fraction that has a negative sign in front of it. For example, \(-\frac{3}{5}\) is a negative fraction.

When you're multiplying negative fractions, it's important to remember that two negatives make a positive. If you multiply \(-\frac{3}{5}\) by another \(-\frac{3}{5}\), the result will be positive because the two negative signs cancel each other out. However, if you have an odd number of negative fractions being multiplied together, like three in this exercise, the result is negative.

This rule stems from the fact that every pair of negative signs cancels each other to create a positive product. When one unpaired negative remains, the entire expression remains negative.

Multiplication of fractions

The multiplication of fractions involves multiplying their numerators together and their denominators together. It's straightforward if you follow this basic rule.

For example, when multiplying the fraction \(\left( -\frac{3}{5} \right)\) by itself, you multiply the top numbers: \(-3 \times -3 = 9\). Then, you multiply the bottom numbers: \(5 \times 5 = 25\). Thus, \(\left( -\frac{3}{5} \right) \times \left( -\frac{3}{5} \right) = \frac{9}{25}\).

If you multiply this result by another fraction \(-\frac{3}{5}\), remember to multiply both the numerators and the denominators again:

• Numerator: \(9 \times -3 = -27\)
• Denominator: \(25 \times 5 = 125\)

This results in \(-\frac{27}{125}\). You see how fractions can be multiplied multiple times like this, easily following the multiply-across strategy.

Cubic powers

Cubic powers refer to raising a number to the power of three. It means you multiply the number by itself two more times. In mathematical terms, if you have \(a^3\), it represents \(a \times a \times a\).

When dealing with fractions, especially negative ones, the same rules apply. For a fraction \(-\frac{3}{5}\) raised to the cubic power, compute it as:

• \((-\frac{3}{5})\times (-\frac{3}{5}) = \frac{9}{25}\)
• Then, \(\frac{9}{25} \times (-\frac{3}{5}) = -\frac{27}{125}\)

This sequence shows that raising a negative fraction to the power of 3 gives a negative result. The negative sign persists because the negative base is paired with an odd exponent. Always remember, raising any negative number to an odd power will yield a negative product.