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Squaring fractions involves a simple yet specific process. When you square a fraction, you square both the numerator and the denominator individually. This means if you have a fraction like \( \left(\frac{a}{b}\right)^2 \), you will do the following calculations:

• Square the numerator: \( a^2 \)
• Square the denominator: \( b^2 \)

This results in the fraction \( \frac{a^2}{b^2} \). For example, in the expression \( \left( -\frac{2}{9} \right)^2 \), the negative sign is squared along with the number, resulting in a positive fraction since \( (-2)^2 = 4 \) and \( 9^2 = 81 \). So, \( \left( -\frac{2}{9} \right)^2 = \frac{4}{81} \).
Remember, squaring a fraction can never yield a negative result because squaring any real number results in a positive value.

Multiplying fractions is straightforward: you multiply the numerators together to get the new numerator and the denominators together for the new denominator. Here’s a step-by-step guide:

• Multiply the numerators: \( a \times c \)
• Multiply the denominators: \( b \times d \)

So, the result of \( \frac{a}{b} \times \frac{c}{d} \) is \( \frac{a \times c}{b \times d} \).
In the example \( \left(\frac{2}{3}\right)\left(\frac{1}{2}\right) \), you multiply the numbers in the numerators, \( 2 \times 1 = 2 \), and the denominators, \( 3 \times 2 = 6 \). Thus, the product is \( \frac{2}{6} \). Simplification follows, by dividing both the numerator and denominator by their greatest common divisor, which in this case is 2, giving \( \frac{1}{3} \).
Multiply first, simplify later – this helps keep track of the fractions correctly.

To add or subtract fractions, they must have the same denominator, known as a "common denominator." Finding a common denominator involves the following steps:

• Identify or calculate the least common multiple (LCM) of the denominators.
• Adjust each fraction so that the denominators are the same.

For example, when adding \( \frac{4}{81} + \frac{1}{3} \), we first determine the LCM of 81 and 3 which is 81. We convert \( \frac{1}{3} \) to an equivalent fraction with a denominator of 81.
Multiply the numerator and the denominator of \( \frac{1}{3} \) by 27 (since \( 3 \times 27 = 81 \)):

• New fraction: \( \frac{1 \times 27}{3 \times 27} = \frac{27}{81} \)

Now the fractions have the same denominator, and we can add them directly: \( \frac{4}{81} + \frac{27}{81} = \frac{31}{81} \).
Remember, when denominators are aligned, addition and subtraction are much simpler. Always take time to find that common denominator for these operations.