Problem 75 - Elementary Algebra

Question

\(\frac{1}{4}-\left(\frac{1}{3}\right)^{2}\)

Step-by-Step Solution

Answer

\( \frac{5}{36} \)

Step 1 - Square the fraction

Calculate \(\frac{1}{3}\)^{2}\. To square a fraction, square both the numerator and the denominator separately. \(\frac{1}{3}\)^{2} = \(\frac{1}{9}\).

Step 2 - Subtract the fractions

To subtract two fractions, find a common denominator. The fractions are \(\frac{1}{4}\) and \(\frac{1}{9}\). The common denominator of 4 and 9 is 36. Convert each fraction: \(\frac{1}{4} = \frac{9}{36}\) and \(\frac{1}{9} = \frac{4}{36}\).

Step 3 - Perform the subtraction

Subtract \(\frac{4}{36}\) from \(\frac{9}{36}\): \(\frac{9}{36} - \frac{4}{36} = \frac{5}{36}\).

Key Concepts

Squaring Fractions

Squaring fractions can sound complicated, but it's quite simple. When we square a fraction, we take both the numerator (the top number) and the denominator (the bottom number) and square them separately. For instance, if we have \(\frac{1}{3}\) and we need to square it, we do the following:

• Square the numerator: 1 squared (\(1^{2}\)) is 1.
• Square the denominator: 3 squared (\(3^{2}\)) is 9.

So, \(\frac{1}{3}\) squared is \(\frac{1^{2}}{3^{2}}\), which equals \(\frac{1}{9}\). Always remember to handle the top and bottom numbers separately. This makes the process straightforward and easy to remember. Squaring fractions is essential before performing more complex operations like subtraction.

Subtracting Fractions

Subtracting fractions involves a few steps to ensure everything is aligned. Before subtracting, both fractions must have the same denominator. Let's see how to do this using our example, \(\frac{1}{4} - \frac{1}{9}\).
We can't directly subtract \(\frac{1}{9}\) from \(\frac{1}{4}\) because the denominators (4 and 9) are different. This brings us to the next step: finding a common denominator.

Finding Common Denominators

Finding a common denominator is the first crucial step when subtracting fractions. The common denominator is the smallest number that both original denominators (in this case, 4 and 9) can divide into evenly. Start by listing multiples of each denominator:

• Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36...
• Multiples of 9: 9, 18, 27, 36...

The smallest common multiple is 36. We now convert each fraction to have 36 as the denominator. To convert:

• For \(\frac{1}{4}\): Multiply both the numerator and denominator by 9 to get \(\frac{9}{36}\).
• For \(\frac{1}{9}\): Multiply both the numerator and denominator by 4 to get \(\frac{4}{36}\).

Now, we have the fractions in a form that allows for easy subtraction: \(\frac{9}{36} - \frac{4}{36}\). Subtracting the numerators, we get \(\frac{5}{36}\). This final fraction, \(\frac{5}{36}\), is the result of our original problem.