Problem 54 - Intermediate Algebra

Question

Evaluate each expression. See Example \(6 .\) $$ \left(-\frac{4}{3}\right)^{3} $$

Step-by-Step Solution

Answer

The result is \( -\frac{64}{27} \).

Step 1: Understand the Power

The expression \( \left(-\frac{4}{3}\right)^{3} \) means we need to multiply \( -\frac{4}{3} \) by itself a total of three times.

Step 2: First Multiplication

First, multiply \( -\frac{4}{3} \) by \( -\frac{4}{3} \). By multiplying fractions, multiply across the numerators and denominators: \[\left( -\frac{4}{3} \right) \times \left( -\frac{4}{3} \right) = \frac{-4 \times -4}{3 \times 3} = \frac{16}{9}\]

Step 3: Second Multiplication

Now take the result from step 2, \( \frac{16}{9} \), and multiply it by \( -\frac{4}{3} \) again: \[\frac{16}{9} \times \left( -\frac{4}{3} \right) = \frac{16 \times -4}{9 \times 3} = \frac{-64}{27}\]

Step 4: Simplify the Result

The fraction \( \frac{-64}{27} \) cannot be simplified further, as 64 and 27 have no common factors other than 1.

Key Concepts

Fraction Multiplication

Multiplying fractions might seem tricky at first, but it becomes straightforward once you understand the basic process.To multiply fractions, you multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. It's as simple as that!
For instance, in our example with \(-\frac{4}{3}\), we encounter this twice:

• The first time, it is multiplied by itself leading to multiplying both the numerators: \(-4 \times -4 = 16\), and the denominators: \(3 \times 3 = 9\), resulting in the fraction \(\frac{16}{9}\).
• The second multiplication involves \(\frac{16}{9}\) by \(-\frac{4}{3}\). Here, \(16 \times -4 = -64\), and \(9 \times 3 = 27\), giving us \(\frac{-64}{27}\).

Remember, always multiply across the top and across the bottom. This rule holds for any fraction multiplication.

Negative Numbers

Handling negative numbers in arithmetic is an essential skill and can be confusing for beginners. Yet, with practice, it becomes intuitive.There are two key rules to remember:

• Multiplying two negative numbers results in a positive number.
• Multiplying a positive number by a negative number results in a negative number.

In our exercise, the first multiplication \(-\frac{4}{3} \times -\frac{4}{3}\) involves two negatives, yielding a positive result, \(\frac{16}{9}\).
In contrast, multiplying \(\frac{16}{9}\) by another negative, \(-\frac{4}{3}\), results in \(\frac{-64}{27}\) since a positive times a negative is negative.Using these simple rules assists in avoiding mistakes, especially under exam conditions.

Simplifying Expressions

Simplifying expressions sometimes allows for cleaner, more manageable results. However, not every expression can be simplified.In our example, the final fraction is \(\frac{-64}{27}\). To determine simplification potential, check for any common factors between the numerator and the denominator:

• The number 64 is composed of the factors: \(2^6\).
• The number 27 is composed of the factors: \(3^3\).

Since these collections of factors have no overlap, other than the trivial factor of 1, \(\frac{-64}{27}\) is indeed in its simplest form.
Being mindful of common factors helps identify if and where simplification can occur, ensuring the expressions are reduced to their most concise format.