Coaching Centre
Index laws
Index notation and the laws for multiplying, dividing and raising powers to powers, including products with coefficients and several pronumerals.
The first three laws
Multiply: add
$$a^m \times a^n = a^{m+n}$$Divide: subtract
$$a^m \div a^n = a^{m-n}$$Power of a power: multiply
$$(a^m)^n = a^{mn}$$Simplify $3a^2b \times 4ab^3$.
Numbers: $3 \times 4 = 12$. $a$: $a^{2+1} = a^3$. $b$: $b^{1+3} = b^4$.
$12a^3b^4$
Simplify $(2x^3)^2$ and $\dfrac{6x^5}{2x^2}$.
$(2x^3)^2 = 4x^6$ (square the $2$ too)
$\dfrac{6x^5}{2x^2} = 3x^3$
$a^5 imes a^3$
$a^8$
$(a^3)^4$
$a^{12}$
$(3x^2)^3$
$27x^6$
Simplify $3a^2b \times 4ab^3$.
Multiply numbers, add indices.
Simplify $(2x^3)^2$.
$2^2 x^{3 \times 2}$.
Simplify $\dfrac{6x^5}{2x^2}$.
$6 \div 2$, $5 - 2$.
Simplify $a^5 \times a^3$.
$5 + 3$.
Simplify $(a^3)^4$.
$3 \times 4$.
Simplify $(3x^2)^3$.
$27x^6$.
Simplify $\dfrac{15m^7}{3m^3}$.
$15 \div 3$, $7 - 3$.
Simplify $2^3 \times 2^4 \times 2^3$ as a power of $2$.
$3 + 4 + 3 = 10$.
New numbers every time. Keep going until you get five in a row.