Year 10AC9M10A01

Index Laws and Rational Indices

Expand, factorise and simplify expressions and solve equations algebraically, applying exponent laws involving products, quotients and powers of variables, and the distributive property.

Four laws cover everything. A fractional index is just a root written another way.

Method

The four laws

a^m a^n = a^{m+n} \quad \frac{a^m}{a^n} = a^{m-n} \quad (a^m)^n = a^{mn} \quad (ab)^n = a^n b^n

Zero index

a^0 = 1 \quad (a \neq 0)

Follows from dividing $a^n$ by itself.

Negative index means reciprocal

a^{-n} = \frac{1}{a^n}

Not a negative answer.

Fractional index means root

a^{1/n} = \sqrt[n]{a} \qquad a^{m/n} = \sqrt[n]{a^m}

Bottom is the root, top is the power.

Worked example

e.g.

\text{Simplify } \dfrac{(2x^3)^2 \times x^{-1}}{x^4} \text{ and evaluate } 27^{2/3}.

Deal with the bracket first. The power hits both parts. $(2x^3)^2 = 4x^6$
Multiply the top terms by adding indices. $4x^6 \times x^{-1} = 4x^5$
Divide by subtracting indices. $\frac{4x^5}{x^4} = 4x$
Now the fractional index. The bottom is the root. $27^{1/3} = 3$
The top is the power. $3^2 = 9$
State the answer. $27^{2/3} = 9$

Practice

Bracket first, then combine indices. Bottom is the root, top is the power.

x^5 \times x^3

Answer

x^8

\frac{y^7}{y^2}

Answer

y^5

(a^4)^3

Answer

a^{12}

16^{1/2}

Answer

4

8^{2/3}

Answer

4

\text{Why does } a^0 = 1?

Answer

\frac{a^n}{a^n} = a^{n-n} = a^0, \text{ and it equals 1.}