Indices · Junior Cert
Indices
Junior Cert Higher · The index laws · Tap NEXT to begin
Section 1 of 5
What Indices Are
An index (or power) is repeated multiplication. In $b^{p}=n$: $b$ is the base, $p$ is the power (index), $n$ is the number (value).
Watch the base
You can only combine powers that have the same base. Different bases must be worked out separately: $2^{3}\times 3^{2} = 8\times 9 = 72$.
Section 2 of 5
Multiply & Divide (same base)
The two laws
$a^{p}\times a^{q} = a^{p+q}$ and $\dfrac{a^{p}}{a^{q}} = a^{p-q}$.
Worked example — multiply
Write $2^{2}\times 2^{3}$ as a single power of $2$.
$2^{2}\times 2^{3} = 2^{2+3}$
$2^{5}$
Worked example — multiply again
Write $3^{3}\times 3^{5}$ as a power of $3$.
$3^{3+5}$
$3^{8}$
Worked example — bigger powers
Write $5^{12}\times 5^{15}$ as a power of $5$.
$5^{12+15}$
$5^{27}$
Worked example — divide
Simplify $\dfrac{5^{4}}{5^{2}}$.
$5^{4-2}$
$5^{2}$
Worked example — divide again
Simplify $\dfrac{2^{50}}{2^{39}}$.
$2^{50-39}$
$2^{11}$
Section 3 of 5
Zero & Negative Powers
Two more laws
$a^{0} = 1$ and $a^{-p} = \dfrac{1}{a^{p}}$.
Worked example — a negative power
Find the value of $5^{-2}$.
$5^{-2} = \dfrac{1}{5^{2}} = \dfrac{1}{25}$
$\dfrac{1}{25}$
Worked example — another
Find the value of $3^{-3}$.
$3^{-3} = \dfrac{1}{3^{3}} = \dfrac{1}{27}$
$\dfrac{1}{27}$
Worked example — one more
Find the value of $6^{-2}$.
$6^{-2} = \dfrac{1}{6^{2}}$
$\dfrac{1}{36}$
Section 4 of 5
Powers & Brackets
Power of a power, and brackets
$(a^{p})^{q} = a^{pq}$ $(ab)^{p} = a^{p}b^{p}$ $\left(\dfrac{a}{b}\right)^{p} = \dfrac{a^{p}}{b^{p}}$.
Worked example — power of a power
Write $(2^{5})^{3}$ as a power of $2$.
$(2^{5})^{3} = 2^{5\times 3}$
$2^{15}$
Worked example — a bracket with a number
Simplify $(3x)^{2}$.
$(3x)^{2} = 3^{2}x^{2}$
$9x^{2}$
Worked example — a bracket cubed
Simplify $(2y)^{3}$.
$(2y)^{3} = 2^{3}y^{3}$
$8y^{3}$
Worked example — a fraction in a bracket
Simplify $\left(\dfrac{4}{5}\right)^{2}$.
$\dfrac{4^{2}}{5^{2}}$
$\dfrac{16}{25}$
Now you try
You try
Simplify $\left(\dfrac{2}{5}\right)^{3}$.
Pen and paper out — try it before you reveal.
$\dfrac{2^{3}}{5^{3}}$
$\dfrac{8}{125}$
$\dfrac{8}{125}$
Section 5 of 5
Fractional Powers & Surds
The link to roots
$a^{\frac{1}{2}} = \sqrt{a}$, so $\sqrt{3} = 3^{\frac{1}{2}}$.
You try
Write $\dfrac{9}{\sqrt{3}}$ in the form $3^{b}$.
Pen and paper out — try it before you reveal.
$\dfrac{9}{\sqrt{3}} = \dfrac{3^{2}}{3^{1/2}} = 3^{2-\frac{1}{2}}$
$3^{\frac{3}{2}}$
$3^{\frac{3}{2}}$
You try
Write $2^{3}\times 2^{7}$ in the form $2^{b}$.
Pen and paper out — try it before you reveal.
$2^{3+7}$
$2^{10}$
$2^{10}$
You try
Write $\dfrac{3^{12}}{3^{6}}$ in the form $3^{b}$.
Pen and paper out — try it before you reveal.
$3^{12-6}$
$3^{6}$
$3^{6}$
More examples
You try
Simplify $a^{3}\cdot a^{4}$.
Add the powers.
$a^{3+4}$
$a^{7}$
You try
Simplify $\dfrac{a^{9}}{a^{7}}$.
Subtract the powers.
$a^{9-7}$
$a^{2}$
You try
Simplify $(a^{3})^{2}$.
Multiply the powers.
$a^{3\times 2}$
$a^{6}$
That’s Indices.
The multiply, divide, power-of-power, zero, negative and fractional laws — every index question is one of these.