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Exponents refer to the process of multiplying a number by itself a specified number of times. They are a shorthand notation for repeated multiplication. For example, \((\frac{1}{2})^3\) is read as "one-half to the power of three," which means multiply \(\frac{1}{2}\) by itself three times: \(\frac{1}{2} \times \frac{1}{2} \times \frac{1}{2}\). To calculate \((\frac{1}{2})^3\), you'll perform

• \(1^3\) for the numerator, since the numerator is 1, and
• \(2^3\) for the denominator, since the denominator is 2.

This equals \(\frac{1}{8}\) because 1 raised to any power remains 1, and 2 cubed is 8.Remember, when dealing with fractions in exponents, apply the power to both the numerator and the denominator separately.

Addition of fractions requires a common denominator to be performed correctly. A denominator is the bottom number in a fraction, which tells you how many equal parts the item is divided into. Only when fractions share a common denominator can you directly add their numerators.For instance, if you want to add \(\frac{3}{10}\) and \(\frac{1}{5}\), you need the same denominator. Multiply the numerator and denominator of \(\frac{1}{5}\) by 2 to get \(\frac{2}{10}\). Now both fractions can be combined:

• \(\frac{3}{10}\) + \(\frac{2}{10}\)

Add the numerators (3 + 2) to get \(\frac{5}{10}\), which simplifies to \(\frac{1}{2}\). Simplifying helps to make a fraction as simple as possible, usually by dividing both the numerator and the denominator by their greatest common divisor.

Much like addition, subtraction of fractions involves combining fractions with a common denominator. First, ensure that the fractions you wish to subtract have the same denominator. In the expression \(\frac{1}{8} + \frac{1}{2} - \frac{5}{2}\), you'll notice different denominators. Convert \(\frac{1}{2}\) and \(\frac{5}{2}\) to fractions with a denominator 8:

• \(\frac{1}{2}\) becomes \(\frac{4}{8}\) and
• \(\frac{5}{2}\) becomes \(\frac{20}{8}\).

Once converted, you can subtract \(\frac{20}{8}\) from the sum of \(\frac{1}{8} + \frac{4}{8}\). The result is \(\frac{-15}{8}\) after you combine the numerators.

Multiplying fractions is straightforward and requires multiplying the numerators and denominators separately. If you need to multiply a fraction by a whole number, think of the whole number as a fraction with a denominator of 1. For example, if you want to multiply 5 by \(\frac{1}{2}\):

• Write 5 as \(\frac{5}{1}\)
• Multiply the numerators: 5 * 1 = 5
• Multiply the denominators: 1 * 2 = 2

This results in \(\frac{5}{2}\). Simplification is not needed here because the fraction is already in its simplest form. Remember, when multiplying fractions, it's not necessary to find a common denominator like in addition or subtraction.