Rules of Indices
What Are Indices?
An index (also called an exponent or power) tells you how many times a number is multiplied by itself.
Example: ( 3^4 )
( 3^4 = 3 times 3 times 3 times 3 )
So, ( 3^4 = 81 )
So, ( 3^4 = 81 )
Law 1: Multiplying Powers
$$ a^m times a^n = a^{m+n} $$
Rule: When multiplying powers with the same base, keep the base and add the indices.
Example: Simplify ( x^2 times x^5 )
Add the powers: ( 2 + 5 = 7 )
Answer: ( x^7 )
Answer: ( x^7 )
Law 2: Dividing Powers
$$ a^m div a^n = a^{m-n} $$
Rule: When dividing powers with the same base, keep the base and subtract the indices.
Example: Simplify ( y^6 div y^2 )
Subtract the powers: ( 6 – 2 = 4 )
Answer: ( y^4 )
Answer: ( y^4 )
Law 3: Power of a Power
$$ (a^m)^n = a^{mn} $$
Rule: When raising a power to another power, multiply the indices.
Example: Simplify ( (x^3)^2 )
Multiply the powers: ( 3 times 2 = 6 )
Answer: ( x^6 )
Answer: ( x^6 )
Law 4: Zero Power
$$ a^0 = 1 quad (a ne 0) $$
Rule: Any non-zero number raised to the power 0 equals 1.
Example: Simplify ( 5^0 )
Answer: ( 1 )
Law 5: Negative Powers
$$ a^{-n} = frac{1}{a^n} $$
Rule: A negative power means take the reciprocal (flip the fraction).
Example 1: Simplify ( 2^{-3} )
( 2^{-3} = frac{1}{2^3} = frac{1}{8} )
Example 2: Simplify ( x^{-4} )
( x^{-4} = frac{1}{x^4} )
Law 6: Fractional Powers
$$ a^{frac{1}{n}} = sqrt[n]{a} $$
Rule: A fractional power represents a root.
Example 1: Simplify ( 9^{frac{1}{2}} )
( 9^{frac{1}{2}} = sqrt{9} = 3 )
Example 2: Simplify ( 27^{frac{1}{3}} )
( 27^{frac{1}{3}} = sqrt[3]{27} = 3 )
Mixed Examples
Example 1: Simplify ( (-4a^{-3})^2 )
Square each part:
( (-4)^2 = 16 )
( (a^{-3})^2 = a^{-6} )
Answer: ( 16a^{-6} )
( (-4)^2 = 16 )
( (a^{-3})^2 = a^{-6} )
Answer: ( 16a^{-6} )
Example 2: Simplify ( (2a)^{-3} )
( (2a)^{-3} = frac{1}{(2a)^3} )
( = frac{1}{8a^3} )
( = frac{1}{8a^3} )
Summary Reminder:
- Add powers when multiplying.
- Subtract powers when dividing.
- Multiply powers when raising a power to a power.
- Negative power = reciprocal.
- Fractional power = root.