Coaching Centre
Zero and negative indices
The zero index, negative indices as reciprocals, and simplifying expressions to positive indices.
The zero index
$\dfrac{a^3}{a^3} = a^{3-3} = a^0$, but anything divided by itself is $1$. So $a^0 = 1$ for any $a \ne 0$.
Negative indices
Each step left halves the value: 2⁰ = 1, 2⁻¹ = ½, 2⁻² = ¼, ...
Evaluate $2^{-3}$ and write $5x^{-2}$ with a positive index.
Evaluate $2^{-3}$.
$\dfrac{1}{2^3}$.
Evaluate $(3x)^0$.
Anything (non-zero) to the power $0$ is $1$.
Evaluate $3x^0$.
Only $x$ is raised to $0$.
Evaluate $10^{-2}$ as a decimal.
$\dfrac{1}{100}$.
Evaluate $4^{-1}$.
Reciprocal of $4$.
Write $5x^{-2}$ with a positive index.
Only $x$ moves.
Simplify $x^2 \div x^5$, leaving a negative index.
$2 - 5 = -3$.
Evaluate $\left(\dfrac{1}{3}\right)^{-2}$.
$3^2$.
New numbers every time. Keep going until you get five in a row.