Year 9 Indices and surds: revision mind map

1Index notation

$2^5$: 2 is the base, 5 is the index (how many 2s multiply).

$2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32$

Prime factors: $72 = 2^3 \times 3^2$

2Multiplying and dividing

Same base: multiply, add the indices. Divide, subtract them.

$a^m \times a^n = a^{m+n}$   $a^m \div a^n = a^{m-n}$

$x^3 \times x^4 = x^7$   $6a^5 \div 2a^2 = 3a^3$

Watch out: $2^3 \times 2^4 = 2^7$, not $4^7$

3Power of a power, zero index

Power of a power: multiply the indices. Anything to the power 0 is 1.

$(a^m)^n = a^{mn}$: $(x^2)^5 = x^{10}$

$7^0 = 1$   $5x^0 = 5$

4Brackets and fractions

The power applies to everything inside the brackets.

$(2x^3)^2 = 4x^6$

$\left(\tfrac{x}{3}\right)^2 = \tfrac{x^2}{9}$

Watch out: square the 2 too: $(2x)^2 = 4x^2$

5Negative indices

A negative index means one over: $a^{-n} = \dfrac{1}{a^n}$.

$2^{-3} = \tfrac18$   $x^{-2} = \tfrac{1}{x^2}$   $\dfrac{1}{a^{-2}} = a^2$

6Scientific notation

Write numbers as $a \times 10^n$ with $1 \le a < 10$.

$45\,000 = 4.5 \times 10^4$

$0.0062 = 6.2 \times 10^{-3}$

Watch out: big numbers get a positive power, small ones a negative power.

7Scientific notation and significant figures

Round the first number to the significant figures asked for.

$0.000\,456\,7$ to 2 s.f.: $4.6 \times 10^{-4}$

$3\,870\,000$ to 2 s.f.: $3.9 \times 10^6$

$(2.4 \times 10^5) \times (3 \times 10^3) = 7.2 \times 10^8$

8Fractional indices and surds

$a^{\frac12} = \sqrt a$, $a^{\frac13} = \sqrt[3]{a}$. A surd is a root that isn't a whole number, like $\sqrt2$.

$9^{\frac12} = 3$   $8^{\frac13} = 2$

Simplify: $\sqrt{12} = \sqrt4 \times \sqrt3 = 2\sqrt3$

9Simple operations with surds

Add or subtract like surds only. Multiply: multiply the numbers outside and inside.

$3\sqrt5 + 2\sqrt5 = 5\sqrt5$   $\sqrt3 \times \sqrt5 = \sqrt{15}$

$2\sqrt3 \times 4\sqrt2 = 8\sqrt6$

$\sqrt8 + \sqrt{18} = 2\sqrt2 + 3\sqrt2 = 5\sqrt2$

Watch out: $\sqrt2 + \sqrt3 \ne \sqrt5$