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Indices

Quick Overview

​Addition & Subtraction!

​LIKE TERMS: like terms, in algebra, are terms which have the same variable. This is important when adding and subtracting in algebraic expressions.

​

Some examples are below:

• ​2a & 8a are LIKE

• ​3a & 7b are UNLIKE

• ​-5c & 3c are LIKE

• ​8yz & 9z are UNLIKE (because '8' has z and y, whilst 9 has only an z)

​Remember that 2a = 2 x a (2 multiplied by 'a')

Product Law

x^m . xⁿ = x^m+n

​

Examples:

2a x 3a² = 6a³

​​Look at this first example - you multiply the '2' and the '3' as normal, then first 'a' is to the power of 1 and the second 'a²' is the power of 2. So 1 + 2 = 3, therefore a³. Put this altogether now, 6a³

​

5b^-3 x 3b² = 15b^-1 = 15/b

​

a^-2b⁴ x a^-4c² = a^-6b⁴c²

​

^{Remember '.' means multiply}

Multiple Choice

Simplify:

$$-3y^4\times4xy^3$$  

• 12xy⁷

• -12xy⁷

• 12x⁷y

• -12x⁷y

Multiple Choice

Simplify: 

$$nm^2\ \times\ 4m$$  

• $$4mn^3$$  

• $$4$$  

• $$4mn$$  

• $$4m^3n$$  

Multiple Choice

Simplify:

$$4xy\ \times\ 3x^3y^2z^4$$  

• $$16xyz$$  

• $$-12x^4y^3z^4$$  

• $$12x^4y^3z^4$$  

• $$16x^4y^3z^4$$  

Quotient Law

x^m/xⁿ = x^m-n

​x^m ÷ xⁿ = x^m-n

Examples:

9a⁴/3a² = 3a²

​For this question 3 goes into 9 three times, so you know the answer will start with a 3. Then looking at the 'a' in the question, 4 minus 2 is two, so 3a² is the answer.

18a²b³/3a⁶b = 3a^-4b² = 3b²/a⁴

​

10cd³/2d³ = 5c

​

​^{Remember when an expression has a negative power (example 2) you can place that part on the bottom of a fraction, removing the negative from the power.}

Multiple Choice

Simplify:

$$\frac{2x^3y^4}{3x^4}$$  

• $$\frac{2y^4}{3x}$$  

• $$6xy$$  

• $$\frac{3y^2}{2x}$$  

• $$2x^5y^4$$  

Multiple Choice

Simplify:

$$\frac{2a^2b^3}{4a^3b^4}$$  

• $$\frac{ab}{2}$$  

• $$\frac{1}{ab}$$  

• $$\frac{1}{2ab}$$  

• $$8ab$$  

Multiple Choice

Simplify:

$$-\frac{mnp^2}{4m^4p^2}$$  

• $$\sqrt{4mn}$$  

• $$-\frac{n}{m^3}$$  

• $$-\frac{n}{4m^3}$$  

• $$4m^3n$$  

Other Laws/Rules

x⁰ = 1

x^-m = 1/x^m

(x^m)ⁿ = x^m.n = x^mxn

Multiple Choice

Simplify:

$$\left(2a^2b^2\right)^2$$  

• $$2a^4b^4$$  

• $$4a^4b^4$$  

• $$-4a^4b^4$$  

• $$4ab$$  

Multiple Choice

Simplify:

$$\left(4x^2y\right)^3$$  

• $$64x^6y^3$$  

• $$16x^6y^3$$  

• $$64x^5y^4$$  

• $$16x^5y^4$$  

Multiple Choice

Solve:

$$x^0$$  

• 1

• 0

• $$x$$  

Multiple Choice

Re-arrange this expression:

  $$3x^2y^{-4}z^3$$  

• $$-3x^2y^{-4}z^3$$  

• $$\frac{3x^2z^3}{y^4}$$  

• $$\frac{3x^2}{y^4z^3}$$  

• $$\frac{3}{x^2y^4z^3}$$  

SUMMARY

Product Law: x^m . xⁿ = x^m+n

​

Quotient Law: x^m/xⁿ = x^m÷ xⁿ = x^m-n

​

Other:

^x⁰ = 1

​

x^-m = 1/x^m

​

(x^m)ⁿ = x^m.n = x^mxn

INDICES

Expressing a power of a product as a product of powers

Multiple Choice

$$\left(3\times a\times b\right)^3$$  Express as a product of powers

• $$28a^3b^3$$  

• $$3^3\times a^3\times b^3$$  

• $$27ab^3$$  

• $$27\left(ab\right)^3$$  

Multiple Choice

$$\left(2\times3\times x\right)^4$$  Express as a product of powers

• $$8\times12\times x^4$$  

• $$3^4\times2^4\times x^2$$  

• $$96x^4$$  

• $$2^4\times3^4\times x^4$$  

a power of a product.

Multiple Choice

$$9x^2$$  express as a power of a product

• $$3^2\times x^2$$  

• $$\left(3x\right)^2$$  

• $$\left(3x^2\right)$$  

• $$9\times x\times x$$  

Multiple Choice

$$64y^3x^6$$  express as a power of a product

• $$8^2\times x^2\times y^3$$  

• $$\left(4xy\right)^3$$  

• $$\left(4yx^2\right)^3$$  

• $$64\times y\times y\times x\times x\times x$$  

The power of a negative integer

Poll

Even power of a negative number is

• Negative

• Positive

Poll

Odd power of a negative number is

• Positive

• Negative

Laws of Indices

Poll

How confident do you feel with this topic now?

• Still have no idea

• Sort of understand

• Pretty Good

• GREAT

Multiple Choice

• 12xy⁷

• -12x⁷y

• -12xy⁷

• 12x⁷y

Multiple Choice

• 4mn³

• 4mn

• -4m³n

• 4m³n

Multiple Choice

• 16xyz

• -12x⁴y³z⁴

• 12x⁴y³z⁴

• 16x⁴y³z⁴

Multiple Choice

• $$\frac{2y^4}{3x}$$

• $$\frac{3y^2}{2x}$$

• 6xy

• $$\frac{2}{3y^5}$$

Multiple Choice

• $$\frac{ab}{2}$$

• $$\frac{1}{2ab}$$

• $$\frac{1}{ab}$$

• 6ab

Multiple Choice

• $$\sqrt{4mn}$$

• 10n

• $$-\frac{n}{4m^3}$$

• $$-\frac{n}{m^3}$$

Multiple Choice

• 2u⁴v⁴

• -4u⁴v⁴

• 4u⁴v⁴

• 4uv

Multiple Choice

• 4x²y⁶

• 4x⁶y²

• 4x^-2y²

• -4x²y²

Multiple Choice

• 128x⁴y⁸z⁴

• -128x⁴y⁴z⁴

• 256x⁴y⁸z⁴

• 256x⁴y⁶z⁴

Multiple Choice

• $$\frac{xy}{4}$$

• $$\frac{1}{xy}$$

• $$\frac{1}{x^4y}$$

• $$\frac{1}{xy^4}$$

Multiple Choice

• a⁹b

• 4ab

• a⁹b⁹

• ab⁹

Multiple Choice

• 4m²n²

• 2m⁴n³

• -2m³n³

• 2m⁴n³

Multiple Choice

• 2x

• $$\frac{1}{2x^2}$$

• $$2x^2$$

• $$\frac{1}{2x}$$

Multiple Choice

• $$x^{\frac{5}{x}}$$

• $$\frac{x^5}{y^4}$$

• $$\frac{y^5}{x^4}$$

• 2xy

Multiple Choice

• $$16mn^5$$

• $$\frac{3m^5}{4n}$$

• $$\frac{4n}{3m^5}$$

• $$4^n$$

Multiple Choice

• $$\frac{xy^9}{64}$$

• $$\frac{x^9y^{15}}{64}$$

• $$64x^9y^{15}$$

• $$64xy^{15}$$

Multiple Choice

• $$2abc$$

• $$2a^2b^5c^{11}$$

• $$\frac{a^{11}c^2}{2b^5}$$

• $$\frac{a^{11}}{bc}$$

Multiple Choice

• $$\frac{1}{4y^3z}$$

• $$4y^{3z}$$

• $$4y^3z$$

• $$\frac{1}{4yz^3}$$

Multiple Choice

• $$\frac{z^2}{x^2y^{10}}$$

• $$\frac{x^2y^2}{z^{10}}$$

• $$4xy$$

• $$xyz^2$$