Learning Objectives
- Simplify fractions raised to a power.
- Simplify square roots of fractions.
Key words
- Base: The number being raised to a power in an exponential expression
- Exponent: The power in an exponential expression
- Principal square root: The positive square root
- Negative square root: The negative square root
Exponents
Exponents are used as a concise way to write multiple multiplication. For example, [latex]2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 2=2^{7}[/latex]. The exponent [latex]7[/latex] tells us how many times we have to multiply the base [latex]2[/latex] by itself.
This same notation is used to raise a fraction to a power.
Examples
Identify the base and the exponent in each term:
1. [latex]\left ( \frac{2}{7}\right )^{6}[/latex] Base = [latex]\frac{2}{7}[/latex] and exponent = [latex]6[/latex]
2. [latex]\left ( \frac{-5}{9}\right )^{4}[/latex] Base = [latex]\frac{-5}{9}[/latex] and exponent = [latex]4[/latex]
3. [latex]\left ( -\frac{1}{25}\right )^{3}[/latex] Base = [latex]-\frac{1}{25}[/latex] and exponent = [latex]3[/latex]
Try It
Identify the base and the exponent in each term:
1. [latex]\left ( \frac{4}{3}\right )^{5}[/latex] Base = [latex]\frac{4}{3}[/latex] and exponent = [latex]5[/latex]
2. [latex]\left ( \frac{-1}{8}\right )^{7}[/latex] Base = [latex]\frac{-1}{8}[/latex] and exponent = [latex]7[/latex]
3. [latex]\left ( -\frac{11}{5}\right )^{3}[/latex] Base = [latex]-\frac{11}{5}[/latex] and exponent = [latex]3[/latex]
Examples
1. [latex]\left(\frac{2}{3}\right )^{2}[/latex] = [latex]\frac{2}{3}\cdot\frac{2}{3}=\frac{4}{9}[/latex]
2. [latex]\left (\frac{-1}{2}\right )^{3}[/latex] = [latex]\frac{-1}{2}\cdot\frac{-1}{2}\cdot\frac{-1}{2}=\frac{-1}{8}[/latex]
3. [latex]\left (-\frac{3}{4}\right )^{4}[/latex] = [latex]-\frac{3}{4}\cdot -\frac{3}{4}\cdot -\frac{3}{4}\cdot -\frac{3}{4}=\frac{81}{64}[/latex]
4. [latex]\left(\frac{1}{2}\right )^{5}=\frac{1}{2}\cdot\frac{1}{2}\cdot\frac{1}{2}\cdot\frac{1}{2}\cdot\frac{1}{2}=\frac{1}{32}[/latex]
5. [latex]\left(\frac{-2}{5}\right )^{3}=\frac{-2}{5}\cdot\frac{-2}{5}\cdot\frac{-2}{5}=\frac{-8}{125}[/latex]
6. [latex]\left(-\frac{2}{7}\right )^{2}=-\frac{2}{7}\cdot -\frac{2}{7}=\frac{4}{49}[/latex]
Try It
Simplify by writing the exponential term as multiple multiplication:
1. [latex]\left(\frac{4}{5}\right )^{2}[/latex]
2. [latex]\left (\frac{-2}{5}\right )^{3}[/latex]
3. [latex]\left (-\frac{2}{3}\right )^{4}[/latex]
Square Roots
The principal square root of a positive integer is the positive number that multiplies by itself to give the integer. As we saw in the section on integers, a positive integer has two square roots. The principal square root is positive and uses the radical sign for notation. For example, [latex]\sqrt{4}=2[/latex]. The integer [latex]4 [/latex] is a perfect square since its square root is a whole number. [latex]4 [/latex] also has a negative square root notated with a negative sign in front of the radical: [latex]-\sqrt{4}=-2[/latex].
To find the square root of a fraction, we look for the fraction that squares to give the original fraction.
For example, to find the square root of [latex]\frac{4}{9}[/latex] we are looking for a fraction that when squared gives us [latex]\frac{4}{9}[/latex]. Since [latex]2^2 = 4[/latex] and [latex]3^{2} =9[/latex], [latex]\frac{2^{2}}{3^{2}}=\frac{4}{9}[/latex]. Therefore, [latex]\sqrt{\frac{4}{9}}=\frac{2}{3}[/latex].
Notice that [latex]\sqrt{4}=2[/latex] and [latex]\sqrt{9}=3[/latex]. This means that [latex]\sqrt{\frac{4}{9}}=\frac{\sqrt{4}}{\sqrt{9}}=\frac{2}{3}[/latex].
SQUARE ROOT RULE FOR FRACTIONS
[latex]\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}[/latex] for any whole numbers [latex]a, b;\; b \ne 0[/latex].
Examples
Simplify:
1. [latex]\sqrt{\frac{25}{16}}=\frac{\sqrt{25}}{\sqrt{16}}=\frac{5}{4}[/latex]
2. [latex]-\sqrt{\frac{81}{4}}=-\frac{\sqrt{81}}{\sqrt{4}}=-\frac{9}{2}[/latex]
3. [latex]\sqrt{\frac{121}{144}}=\frac{\sqrt{121}}{\sqrt{144}}=\frac{11}{12}[/latex]
4. [latex]\sqrt{\frac{-9}{25}}=\frac{\sqrt{-9}}{\sqrt{25}}[/latex] is undefined in the set of real numbers
Try It
Simplify:
1. [latex]\sqrt{\frac{36}{49}}[/latex]
2. [latex]\sqrt{\frac{100}{25}}[/latex]
3. [latex]-\sqrt{\frac{16}{121}}[/latex]
4. [latex]\sqrt{\frac{-16}{81}}[/latex]
