Indices
Before reading this guide, it may be useful to look at M7 Indices.
Rules of Indices when working with the same base
| RULE | ALGEBRAIC RULE | NUMERICAL EXAMPLE |
|---|---|---|
| 1. When multiplying, ADD the indices | \(a^3 \times a^2 = a^5\) | \(2^3 \times 2^2 = 2^5 = 32\) |
| 2. When dividing, SUBTRACT the indices | \(a^5 梅 a^2 = a^3\) | \(3^5 梅3^2 = 3^3 = 27\) |
| 3. When the power is raised to another power, MULTIPLY the indices. | \((a^3)^2 = a^6\) | \((3^3)^2 = 3^6 = 729\) |
| 4. Anything to the power of zero is 1 | \(a^0 = 1\) | \(43^0 = 1\) |
| 5. Negative indices indicate a fraction. | \(a^{-n} = \frac{1}{a^n}\) | \(3^{-2} = \frac{1}{3^2} = \frac{1}{9}\) |
We can add a further rule for fractional indices -
| RULE | ALGEBRAIC RULE | NUMERICAL EXAMPLE |
|---|---|---|
| 1. When multiplying, ADD the indices | \(a^3 \times a^2 = a^5\) | \(2^3 \times 2^2 = 2^5 = 32\) |
| 2. When dividing, SUBTRACT the indices | \(a^5 梅 a^2 = a^3\) | \(3^5 梅3^2 = 3^3 = 27\) |
| 3. When the power is raised to another power, MULTIPLY the indices. | \((a^3)^2 = a^6\) | \((3^3)^2 = 3^6 = 729\) |
| 4. Anything to the power of zero is 1 | \(a^0 = 1\) | \(43^0 = 1\) |
| 5. Negative indices indicate a fraction. | \(a^{-n} = \frac{1}{a^n}\) | \(3^{-2} = \frac{1}{3^2} = \frac{1}{9}\) |
| 6. Fractional indices 鈥冣赌Numerator indicates a power 鈥冣赌Denominator indicates a root | \(a^{\frac{b}{c}}=\sqrt[c]{a^b}\) | \(8^{\frac{2}{3}}=(\sqrt[3]{8})^2 = 2^2 = 4\) |
Evaluating fractional indices
Example
Evaluate \( 25^{\frac{1}{2}}\)
Solution:
\(25^{\frac{1}{2}} = \sqrt{25} = 5\)
Example
Evaluate \(81^{\frac{1}{4}}\times 9^{-1}\)
Solution
\( 81^{\frac{1}{4}} = \sqrt[4]{81} = 3\)
\( 9^{-1} = \frac{1}{9}\)
\(81^{\frac{1}{4}} \times 9^{-1} = 3 \times \frac{1}{9} = \frac{3}{9} = \frac{1}{3}\)
Answer
\(\frac{1}{3}\)
Example
Evaluate \(100^{鈥揬frac{3}{2}}\)
Solution
\(100^{鈥揬frac{3}{2}} = \frac{1}{100^{\frac{3}{2}}} = \frac{1}{\sqrt{(100)}^3} = \frac{1}{1000}\)
Question
Evaluate \(\frac {1}{16^{\frac{3}{4}}}\times 2^{-3}\)
Solution
\(\frac{1}{16^{\frac{3}{4}}} = \frac{1}{(\sqrt[4]{16})^3} = \frac{1}{(2)^3} = \frac{1}{8}\)
\(2^{-3} = \frac{1}{(2)^3} = \frac{1}{8}\)
\(\frac{1}{16^{\frac{3}{4}}} \times 2^{-3} = \frac{1}{8} \times \frac{1}{8} = \frac{1}{64} \)
Simplifying expressions with fractional indices
Example
Simplify \((a^4)^{鈥揬frac{1}{2}} \times a^{\frac{3}{2}} \)
Solution
\((a^4)^{\mathbf{鈥揬frac{1}{2}}}\times a^{\frac{3}{2}}\)
Use rule 3 鈥 multiply indices to raise a further power
\((a^{\mathbf{4}})^{\mathbf {鈥揬frac{1}{2}}}\times a^{\frac{3}{2}} = a^{\mathbf{-2}} \times a^{\mathbf{\frac{3}{2}}}\)
Now use rule 1 鈥 add the indices (\(= -2 + \frac{3}{2} = -{\frac{1}{2}}\))
\(= a^{鈥揬frac{1}{2}}\)
Question
Simplify \(t^{鈥揬frac{1}{4}} \times t^0 \times t^\frac{3}{4}\)
Solution
\(t^{鈥揬frac{1}{4}} \times t^0 \times t^\frac{3}{4}\)
Use Rule 1: add the indices
\( = t^{({鈥揬frac{1}{4} + 0 + \frac {3}{4}})}\)
\( = t^{\frac{1}{2}}\)
Solving Equations using Rules for indices
Example
Solve the equation \( 4^{-y} = \sqrt{16^3}\)
Solution
Rewrite Right Hand Side (RHS) in powers of 4
\(\sqrt{16^3} = 4^3\)
Equate Right Hand Side and Left Hand Side (LHS)
\(4^{-y} = 4^3\)
Answer: \(\mathbf {y = -3}\)
Question
Solve the equation \(0.01^y = 1000^{\frac {-1}{3}}\)
Solution
Rewrite both sides of the equation as powers of 10
LHS
\(0.01^y = (\frac{1}{100})^y = (\frac{1}{10^2})^y = \frac{1}{10^{2y}}\)
RHS
\(1000^{\frac{-1}{3}} = \frac{1}{\sqrt[3]{1000}} = \frac{1}{10}\)
Equate sides
\(\frac{1}{10^{2y}} = \frac{1}{10}\)
Answer:
2y = 1
y = 0.5
Test yourself
More on M8: Algebra
Find out more by working through a topic
- count4 of 7
