Laws of Indices in Algebra

Count how many times m is multiplied by itself.

When you multiply powers of the same letter, add the powers.

When you divide powers of the same letter, subtract the powers.

A power of a power means you multiply the powers.

A letter on its own has a power of 1.

Think about x^3\div x^3: subtract the powers, and also work out what it must equal.

Part (a)

1. x^3\div x^3=x^{3-3}=x^0, and anything divided by itself is 1.
2. So x^0=1.

Part (b)

1. y^0=1, so 7y^0=7\times1=7.
2. Only the y is raised to the power 0, not the 7.

Multiply the numbers, then add the powers of x.

1. Numbers: 3\times5=15.
2. Letters: x^4\times x^2=x^{4+2}=x^6.
3. So 3x^4\times5x^2=15x^6.

Divide the numbers, then subtract the powers of y.

1. Numbers: 24\div6=4.
2. Letters: y^9\div y^3=y^{9-3}=y^6.
3. So 24y^9\div6y^3=4y^6.

Deal with the numbers, the as and the bs separately. A letter with no power shown has a power of 1.

1. Numbers: 3\times6=18.
2. a^2\times a=a^{2+1}=a^3.
3. b\times b^3=b^{1+3}=b^4.
4. So the answer is 18a^3b^4.

Everything inside the bracket is squared, the 3 as well as the x^4.

1. (3x^4)^2=3x^4\times3x^4.
2. 3\times3=9 and x^4\times x^4=x^8.
3. So (3x^4)^2=9x^8.
4. It is not 3x^8: the 3 must be squared too.

Subtract the powers, even though the answer will be negative.

1. w^2\div w^6=w^{2-6}.
2. 2-6=-4, so the answer is w^{-4}.

Simplify the top first, then divide.

1. Top: a^4\times a^7=a^{4+7}=a^{11}.
2. Then a^{11}\div a^5=a^{11-5}=a^6.

For a power of a power, you multiply the powers.

1. (x^n)^3=x^{3n}.
2. 3n=18, so n=6.
3. Check: (x^6)^3=x^{6\times3}=x^{18}.

Squaring means multiplying the whole term by itself, and a negative times a negative is positive.

1. (-4y^3)^2=(-4y^3)\times(-4y^3).
2. -4\times(-4)=16 and y^3\times y^3=y^6.
3. So (-4y^3)^2=16y^6.

Is Jake adding the terms or multiplying them?

Part (a)

1. p^4+p^4 is two lots of p^4, an addition.
2. You add the powers only when you multiply, as in p^4\times p^4=p^8.

Part (b)

1. p^4+p^4 is 1 lot of p^4 add 1 lot of p^4.
2. So p^4+p^4=2p^4.

Simplify the top first: multiply the numbers and add the powers. Then divide.

1. Top: 8\times3=24 and a^6\times a^2=a^8, so the top is 24a^8.
2. Divide the numbers: 24\div6=4.
3. Subtract the powers: a^8\div a^5=a^3.
4. So the answer is 4a^3.

For a power of a power, multiply the powers, and take care with the signs.

Part (a)

1. Multiply the powers: 3\times(-2)=-6.
2. (t^3)^{-2}=t^{-6}.

Part (b)

1. Multiply the powers: -5\times(-2)=10, because a negative times a negative is positive.
2. (t^{-5})^{-2}=t^{10}.

Cube each part inside the bracket: the 2, the a^3 and the b^2.

1. 2^3=2\times2\times2=8.
2. (a^3)^3=a^{3\times3}=a^9.
3. (b^2)^3=b^{2\times3}=b^6.
4. So (2a^3b^2)^3=8a^9b^6.

Volume of a cuboid = length \times width \times height.

Part (a)

1. Volume =2x\times3x^2\times x^4.
2. Numbers: 2\times3=6. Letters: x\times x^2\times x^4=x^{1+2+4}=x^7.
3. Volume =6x^7 cm³.

Part (b)

1. Area of the base =3x\times x^2=3x^3.
2. Height = volume \div area of base =6x^7\div3x^3.
3. 6\div3=2 and x^7\div x^3=x^4, so the height is 2x^4 cm.

The first statement tells you what a and b add to; the second tells you what one minus the other is.

Part (a)

1. Multiplying adds the powers, so a+b=9.
2. Dividing subtracts the powers, so a-b=3.
3. Two numbers that add to 9 and differ by 3 are 6 and 3.
4. So a=6.

Part (b)

1. a+b=9 and a=6, so b=3.
2. Check: x^6\times x^3=x^9 and x^6\div x^3=x^3.