72 total
By the end of this topic, you should be able to:
An index (also called a power or exponent) tells you how many times to multiply a number by itself. The plural of index is indices.
For example, in the expression 2⁴:
Another example: x⁵ means x × x × x × x × x (you multiply x by itself 5 times).
The laws of indices are rules that help you simplify expressions involving powers. These rules only work when you are multiplying or dividing terms – they do NOT work for addition or subtraction.
When you multiply two powers with the same base, you add the indices.
Formula: aᵐ × aⁿ = a⁽ᵐ⁺ⁿ⁾
Example 1: x⁷ × x¹² = x⁽⁷⁺¹²⁾ = x¹⁹
Example 2: 2³ × 2⁵ = 2⁽³⁺⁵⁾ = 2⁸ = 256
Why it works: If you write it out fully, x⁷ × x¹² means (x × x × x × x × x × x × x) × (x × x × x × x × x × x × x × x × x × x × x × x). When you count all the x's, you get 19 of them, which is x¹⁹.
When you divide two powers with the same base, you subtract the indices.
Formula: aᵐ ÷ aⁿ = a⁽ᵐ⁻ⁿ⁾
Example 1: x¹⁰ ÷ x⁴ = x⁽¹⁰⁻⁴⁾ = x⁶
Example 2: 10¹² ÷ 10⁴ = 10⁽¹²⁻⁴⁾ = 10⁸
Why it works: If you write d⁵ ÷ d², you have (d × d × d × d × d) ÷ (d × d). The two d's on top cancel with the two d's on the bottom, leaving you with d × d × d = d³.
When you raise a power to another power, you multiply the indices.
Formula: (aᵐ)ⁿ = aᵐⁿ
Example 1: (x³)² = x⁽³ˣ²⁾ = x⁶
Example 2: (e⁵)⁵ = e⁽⁵ˣ⁵⁾ = e²⁵
Why it works: (x³)² means you're squaring x³, so you get (x × x × x) × (x × x × x) = x⁶.
When you raise a product (two or more numbers multiplied together) to a power, you raise each number to that power separately.
Formula: (ab)ⁿ = aⁿ × bⁿ
Example: (2x)³ = 2³ × x³ = 8x³
Another example: (fg)² = f² × g² = f²g²
When you raise a fraction to a power, you raise both the numerator (top) and denominator (bottom) to that power.
Formula: (a/b)ⁿ = aⁿ/bⁿ
Example: (h/i)² = h²/i²
Another example: (3/x)² = 3²/x² = 9/x²
Sign in to view full notes