Coaching Centre
Index laws
Index laws for multiplying and dividing powers with the same base, the zero index, and raising a power to a power, including coefficients.
The index laws
Simplify $(x^4)^3$ and $(2y^3)^2$.
$(x^4)^3 = x^{12}$
$(2y^3)^2 = 2^2 y^6 = 4y^6$
Simplify $\dfrac{6a^5b^2 \times 2ab^3}{4a^3b^5}$.
Numerator: $12a^6b^5$.
$\dfrac{12a^6b^5}{4a^3b^5} = 3a^3b^0 = 3a^3$
$(a^m)^n$
$a^{mn}$
$(3x)^2$
$9x^2$
$5x^0$
$5$ (only $x$ is raised to $0$)
$(x^4)^3 = x^{?}$. Enter the index.
$4 \times 3$.
Simplify $(2y^3)^2$.
Square the $2$ too.
Simplify $\dfrac{6a^5b^2 \times 2ab^3}{4a^3b^5}$.
The $b$s cancel completely.
Simplify $5x^0$.
$5 \times 1$.
Simplify $(5x)^0$.
Everything in the bracket is raised to $0$.
Simplify $3m^4 \times 4m^2$.
Multiply numbers, add indices.
Simplify $\dfrac{20p^7}{5p^3}$.
Divide numbers, subtract indices.
Evaluate $(2^3)^2$.
$2^6$.
New numbers every time. Keep going until you get five in a row.