Indices — Rules & Evaluating

Section 1 of 3

The Language of Indices

Base, power, value

An index (or power) tells you how many times to multiply. In $b^p = n$: $b$ is the base, $p$ is the power (index) and $n$ is the value. Indices are also called exponentials.

The rules

$a^p \cdot a^q = a^{p+q}$ (multiply → add powers). $\dfrac{a^p}{a^q} = a^{p-q}$ (divide → subtract). $(a^p)^q = a^{pq}$ (power of a power → multiply).

Roots are fractional powers

$a^{\frac{1}{p}} = \sqrt[p]{a}$. In particular $\sqrt{x} = x^{\frac{1}{2}}$ — learn this one.

Section 2 of 3

Evaluating

Worked example — powers and roots

Find the value of (i) $2^3$; (ii) $3^2$; (iii) $16^{\frac{1}{2}}$; (iv) $100^{\frac{1}{2}}$; (v) $5^2$.

(i) $2^3 = 2\times2\times2 = 8$

(ii) $3^2 = 9$

(iii) $16^{\frac{1}{2}} = \sqrt{16} = 4$

(iv) $100^{\frac{1}{2}} = \sqrt{100} = 10$

(v) $5^2 = 25$

$8,\ 9,\ 4,\ 10,\ 25$

Section 3 of 3

Simplifying

Worked example — using the rules

Simplify (i) $(a^2)^4$; (ii) $a^2(a^3)^4$; (iii) $\dfrac{(k^3)^2}{k^4}$.

(i) $(a^2)^4 = a^{2\times4} = a^8$

(ii) $a^2(a^3)^4 = a^2 \cdot a^{12} = a^{14}$

(iii) $\dfrac{(k^3)^2}{k^4} = \dfrac{k^6}{k^4} = k^{2}$

$a^8,\ a^{14},\ k^2$

You try

Simplify $(b^3)^6$.

Multiply the powers.

$(b^3)^6 = b^{18}$

$b^{18}$

You try

Write $(5^2)^{x+1}$ as a single power of $5$.

$(a^p)^q = a^{pq}$.

$(5^2)^{x+1} = 5^{2(x+1)} = 5^{2x+2}$

$5^{2x+2}$

You try

Write $27^{2x-4}$ as a power of $3$.

$27 = 3^3$.

$27^{2x-4} = (3^3)^{2x-4} = 3^{6x-12}$

$3^{6x-12}$

You try

Write $36^{x-3}$ as a power of $6$.

$36 = 6^2$.

$36^{x-3} = (6^2)^{x-3} = 6^{2x-6}$

$6^{2x-6}$

That’s Class 9.

The index rules, evaluating powers and roots, and simplifying. Class 10: solving index equations.