Laws of Indices in Algebra

A power is a short way to write a letter multiplied by itself. x\times x\times x\times x is written x^4, and said “x to the power 4”. The small raised number is the index. The plural of index is indices.

x^2 is said “x squared” and x^3 is said “x cubed”. A letter on its own has index 1, so x is the same as x^1

Don't mix up powers and adding. x^4 means four x's multiplied together, but 4x means four x's added: x+x+x+x=4x

In the exam you are usually asked to simplify an expression such as n\times n\times n, y^9\div y^4 or (x^3)^4, for 1 mark each. Questions with numbers in front of the letters, or with two letters, such as 2x^2y\times5xy^3, are usually worth 2 marks. A power of a power with a negative index, such as (x^4)^{-2}, can come near the end of the paper.

Write out k^3\times k^2 in full: (k\times k\times k)\times(k\times k). That is five k's multiplied together, so k^3\times k^2=k^5

When you multiply powers of the same letter, add the indices:

x^m\times x^n=x^{m+n}

Here m and n stand for the two indices.

The rule only works for the same letter. x^2\times y^3 can't be simplified: it is just written x^2y^3

Write m^6\div m^2 as a fraction, \frac{m\times m\times m\times m\times m\times m}{m\times m}. The two m's on the bottom cancel with two on the top, which leaves four m's. So m^6\div m^2=m^4

When you divide powers of the same letter, subtract the indices:

x^m\div x^n=x^{m-n}

Take the second index away from the first. Never divide the indices: m^6\div m^2 is m^4, not m^3

Subtracting can give the power zero. x^5\div x^5=x^0, but anything divided by itself is 1. So

x^0=1

Any letter to the power zero is 1, so 9y^0=9\times1=9

(x^3)^2 means x^3 squared, which is x^3\times x^3. Adding the indices gives x^6, and 3\times2=6 too.

When a power is raised to another power, multiply the indices:

(x^m)^n=x^{m\times n}

So (x^5)^3=x^{15}, not x^8

A number inside the bracket is raised to the power as well. (2x^3)^2=2x^3\times2x^3=4x^6, because 2^2=4

Sometimes an index is missing. If x^n\times x^4=x^{11}, the indices add to 11, so n+4=11 and n=7. If (x^n)^5=x^{20}, then 5n=20 and n=4

The number in front of a letter is its coefficient. In 3x^2 the coefficient is 3 and the index is 2. Deal with the numbers and the letters separately.

To multiply, multiply the numbers and add the indices:

3x^2\times4x^5=12x^7

because 3\times4=12 and 2+5=7

To divide, divide the numbers and subtract the indices:

12x^6\div3x^2=4x^4

because 12\div3=4 and 6-2=4

With two letters, deal with each letter on its own, and remember that y means y^1. So 2x^2y\times5xy^3=10x^3y^4

Adding is different. h^5+h^5 is two lots of h^5, so it is 2h^5, not h^{10}. You only add the indices when you multiply.

Subtracting a bigger index gives a negative index. m^3\div m^7=m^{3-7}=m^{-4}

Written as a fraction, three m's on top cancel with three of the seven on the bottom. That leaves \frac{1}{m^4}, so m^{-4} means \frac{1}{m^4}. In the exam you can leave a negative index as it is.

All three rules work in the same way with negative indices. Just take care with the arithmetic: m^{-3}\times m^5=m^2, because -3+5=2, and (m^{-2})^4=m^{-8}, because -2\times4=-8

• Mixing up 4m and m^4. m\times m\times m\times m is m^4, but m+m+m+m is 4m
• Dividing the indices. y^{10}\div y^5 is y^5, because you subtract the indices: 10-5=5
• Adding the indices in a power of a power. (x^2)^6 is x^{12}, because you multiply the indices.
• Adding the numbers in front. 2x^3\times3x^4 is 6x^7: multiply the numbers, then add the indices.
• Adding the indices when you add terms. n^3+n^3 is 2n^3, not n^6
• Slips with negative numbers. x^{-4}\times x^6 is x^2, because -4+6=2