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Exponentiating fractions involves raising a fraction to a certain power. It is similar to exponentiating whole numbers, but here you multiply the fraction by itself a number of times equal to the exponent. For example, taking the power of \(\left(\frac{1}{2}\right)^2\), means multiplying \(\frac{1}{2}\) by itself:

• \(\left(\frac{1}{2}\right)^2 = \frac{1}{2} \times \frac{1}{2}\)
• This simplifies to \(\frac{1}{4}\) because both numerators and denominators are multiplied respectively.

Keep in mind:

• To exponentiate a fraction, apply the power to both numerator and denominator.
• Fraction \(\left(\frac{a}{b}\right)^{n}\) becomes \(\frac{a^n}{b^n}\).

This concept is vital to recognize patterns and simplify expressions similar to simple exponents.

Converting a decimal like 0.125 into a fraction involves understanding decimal place value. A decimal can be viewed as a fraction where the denominator is a power of ten. Let's take a look at how to transform 0.125 into a fraction:1. Recognize the number of decimal places: 0.125 has three decimal places. - This means the fraction's denominator will be \(10^3 = 1000\).2. Express the decimal as a fraction: - \(0.125 = \frac{125}{1000}\).3. Simplify the fraction by finding the greatest common divisor (GCD) of 125 and 1000, which is 125. - Divide both the numerator and the denominator by their GCD:\[\frac{125}{1000} = \frac{125 \div 125}{1000 \div 125} = \frac{1}{8}.\]This method ensures that any decimal is correctly converted into a simplified fraction.

When dealing with addition or subtraction of fractions, it’s essential to have a common denominator. The common denominator allows you to combine fractions into one single fraction easily. Here's how to adjust fractions to have a common denominator:To subtract \(\frac{1}{8}\) from \(\frac{1}{4}\):

• The denominators here are 8 and 4. The least common denominator (LCD) is 8 since it is the smallest number into which both denominators can divide without leaving a remainder.
• Convert \(\frac{1}{4}\) to a fraction with the denominator 8:
  • \(\frac{1}{4} = \frac{2}{8}\), achieved by multiplying both the numerator and the denominator of \(\frac{1}{4}\) by 2.

Now you can subtract:\[\frac{2}{8} - \frac{1}{8} = \frac{1}{8}.\]Finding a common denominator keeps fraction operations clear and simplified, aiding in achieving the correct result.