11D

The index laws and the zero index

The first index laws: multiplying and dividing powers with the same base, and why any non-zero number to the power zero equals one.

Two index laws

Multiplying powers

$$a^m \times a^n = a^{m + n}$$

Dividing powers

$$a^m \div a^n = a^{m - n}$$

Example 1●

Simplify (a) $5^4 \times 5^3$ and (b) $x^9 \div x^4$.

(a) $5^{4 + 3} = 5^7$

(b) $x^{9 - 4} = x^5$

Why? $\dfrac{a^3}{a^3} = 1$, but by the division law it is also $a^{3 - 3} = a^0$.

Example 2●●

Simplify $\dfrac{4m^6 \times m^2}{2m^8}$.

Numbers: $4 \div 2 = 2$. Powers: $m^{6 + 2 - 8} = m^0 = 1$.

$= 2$

Flashcards

$a^m imes a^n$

$a^{m + n}$

$a^m div a^n$

$a^{m - n}$

$a^0$

$1$ (for $a \ne 0$)

Practice