Negative Indices & Reciprocals
Negative Indices & Reciprocals
Grade 11 Mathematics • G.C.E. O/L Sri Lanka Local Syllabus
Learning Objectives
- Understand the meaning of a negative index as the reciprocal of a positive power.
- Rewrite algebraic expressions with negative indices using positive indices only.
- Evaluate numerical expressions and fractions raised to negative powers.
Negative Index Rules
Using the quotient law, \(a^0 \div a^n = a^{0-n} = a^{-n}\). Since \(a^0 = 1\), we obtain \(\frac{1}{a^n} = a^{-n}\).
Step-by-Step Worked Examples
- Invert the fraction \(\frac{2}{3}\) to change \(-3\) to \(+3\), and write \(2^{-2}\) as \(\frac{1}{2^2}\): \[\left(\frac{3}{2}\right)^3 \times \frac{1}{2^2}\]
- Expand the powers and multiply: \[\frac{27}{8} \times \frac{1}{4} = \frac{27}{32}\]
- Simplify the numerical coefficient \(\frac{3}{6} = \frac{1}{2}\) and shift terms with negative exponents across the fraction bar: \[\frac{y^2 \cdot y^3}{2 \cdot x^2 \cdot x^4}\]
- Apply the product law in the numerator and denominator: \[\frac{y^{2+3}}{2x^{2+4}} = \frac{y^5}{2x^6}\]
Practice Drill
Show Step-by-Step Answer
Step 1: Divide coefficients (\(\frac{4}{2} = 2\)) and shift negative powers across the fraction line:
\(\frac{2y^3 \cdot y^4}{x^3 \cdot x^2}\)
Step 2: Apply the product law:
\(\mathbf{\frac{2y^7}{x^5}}\)
Show Step-by-Step Answer
Step 1: Convert each term to positive index form:
\(5^{-2} = \frac{1}{5^2} = \frac{1}{25}\) and \(\left(\frac{1}{5}\right)^{-1} = 5^1 = 5\)
Step 2: Add the values:
\(\frac{1}{25} + 5 = \mathbf{5\frac{1}{25}}\) (or \(\frac{126}{25}\))
Show Step-by-Step Answer
Step 1: Simplify inside the bracket first:
\(2^{-1} \times 4^{-1} = \frac{1}{2} \times \frac{1}{4} = \frac{1}{8}\)
Step 2: Apply the outer power \(-2\):
\(\left(\frac{1}{8}\right)^{-2} = 8^2 = \mathbf{64}\)