Problem 9 - Beginning and Intermediate Algebra

Question

Evaluate. \(\left(\frac{1}{5}\right)^{3}\)

Step-by-Step Solution

Answer

\( \left( \frac{1}{5} \right)^{3} = \frac{1}{125} \).

Step 1: Understand the Expression

The expression \( \left( \frac{1}{5} \right)^{3} \) means we need to multiply \( \frac{1}{5} \) by itself a total of three times. This involves repeated multiplication of the fraction, which raises it to the power of 3.

Step 2: Multiply the Fraction by Itself

To simplify \( \left( \frac{1}{5} \right)^{3} \), calculate \( \frac{1}{5} \times \frac{1}{5} \times \frac{1}{5} \). Start by multiplying the first two fractions: \( \frac{1}{5} \times \frac{1}{5} = \frac{1 \times 1}{5 \times 5} = \frac{1}{25} \).

Step 3: Multiply the Result by the Fraction Again

Now, take the result from Step 2 and multiply it by \( \frac{1}{5} \) again: \( \frac{1}{25} \times \frac{1}{5} = \frac{1 \times 1}{25 \times 5} = \frac{1}{125} \). This is the final answer.

Key Concepts

Fractions

A fraction represents a part of a whole and is expressed as \( \frac{a}{b} \), where \( a \) is the numerator, and \( b \) is the denominator.
The numerator represents how many parts we have, whereas the denominator tells us into how many equal parts the whole is divided. In general, understanding fractions is key because they allow us to work with quantities smaller than the whole. Fractions are everywhere

• in cooking (e.g., \( \frac{1}{2} \) cup of sugar),
• in shopping (e.g., \( \frac{3}{4} \) off the price),
• and in many other daily activities.

The specific fraction \( \frac{1}{5} \) in this exercise means that something is divided into 5 equal parts and we're considering 1 part of these 5.
It is crucial to think of fractions as numbers that can be added, subtracted, multiplied, and divided, just like whole numbers.

Powers and Roots

Powers refer to the operation of multiplying a number by itself a certain number of times. When we see \( x^n \), it means we multiply \( x \) by itself \( n \) times.
For the problem \( \left(\frac{1}{5}\right)^3 \), "\( 3 \)" is the power. Powers tell us to perform repeated multiplication:

• In this case: \( \frac{1}{5} \times \frac{1}{5} \times \frac{1}{5} \).
• Each time the fraction is involved in a multiplication, its denominator increases exponentially.

Roots, on the other hand, are essentially the "opposite" of powers, helping us find a number that, when raised to a certain power, gives us the original number.
While roots are not required in this specific problem, they are a related concept when working with powers and are essential for understanding how powers can be undone.

Multiplication of Fractions

When multiplying fractions, the process involves multiplying the numerators together and the denominators together.
This is expressed in the formula:\[\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\]For our exercise, this means:

• First, multiply \( \frac{1}{5} \times \frac{1}{5} \), resulting in \( \frac{1 \times 1}{5 \times 5} = \frac{1}{25} \).
• Then multiply \( \frac{1}{25} \times \frac{1}{5} \), leading to \( \frac{1 \times 1}{25 \times 5} = \frac{1}{125} \).

The multiplication of fractions does not involve finding a common denominator, unlike addition and subtraction.
Instead, it's more straightforward, making it sometimes easier for students when working with fractions.