Fractional Indices (Roots & Powers)
Fractional Indices (Roots & Powers)
Grade 11 Mathematics • G.C.E. O/L Sri Lanka Local Syllabus
Learning Objectives
- Interpret unit fractional exponents (\(a^{\frac{1}{n}}\)) as \(n^{\text{th}}\) roots (\(\sqrt[n]{a}\)).
- Evaluate general fractional powers (\(a^{\frac{m}{n}}\)) and negative fractional powers (\(a^{-\frac{m}{n}}\)).
- Use prime factorization to simplify fractional exponents without a calculator.
Fractional Index Rules
Step-by-Step Worked Examples
- Write \(27\) as a power of its prime factor (\(27 = 3^3\)): \[(3^3)^{\frac{2}{3}}\]
- Multiply the exponents using \((a^m)^n = a^{mn}\): \[3^{3 \times \frac{2}{3}} = 3^2 = 9\]
- Apply the negative index rule to invert the fraction: \[\left(\frac{27}{8}\right)^{\frac{2}{3}}\]
- Write numerator and denominator in prime index form (\(27 = 3^3\) and \(8 = 2^3\)): \[\left(\frac{3^3}{2^3}\right)^{\frac{2}{3}}\]
- Distribute the fractional exponent \(\frac{2}{3}\) to both numerator and denominator: \[\frac{3^{3 \times \frac{2}{3}}}{2^{3 \times \frac{2}{3}}} = \frac{3^2}{2^2} = \frac{9}{4} = 2\frac{1}{4}\]
Practice Drill
Show Step-by-Step Answer
Step 1: Express \(32\) in prime index form (\(32 = 2^5\)):
\((2^5)^{\frac{3}{5}}\)
Step 2: Multiply the exponents:
\(2^{5 \times \frac{3}{5}} = 2^3 = \mathbf{8}\)
Show Step-by-Step Answer
Step 1: Express \(16\) and \(81\) as prime powers (\(16 = 2^4\) and \(81 = 3^4\)):
\((2^4)^{-\frac{3}{4}} \times (3^4)^{\frac{1}{4}}\)
Step 2: Multiply exponents:
\(2^{-3} \times 3^1 = \frac{1}{2^3} \times 3 = \frac{1}{8} \times 3 = \mathbf{\frac{3}{8}}\)
Show Step-by-Step Answer
Step 1: Invert the fraction to remove the negative sign:
\(\left(\frac{49}{25}\right)^{\frac{1}{2}} = \sqrt{\frac{49}{25}}\)
Step 2: Take the square root of numerator and denominator:
\(\frac{7}{5} = \mathbf{1\frac{2}{5}}\)