# Multiplication, division and powers of exponents

Last updated: 11 Nov 2008

Often when a mathematical expression contains more than one exponent it is possible to simplify it.

### Multiplication of exponents

If you multiply two exponents with the **same base** then you simply have to add two exponents. for example,

5 times itself twice times 5 times itself 4 times is 5 times itself 6 times.

We can generalise this to

### Division of exponents

You can do a very similar operation to simplify the division of exponents that have the **same base**. This time, instead of adding the two exponents, you subtract them. For example,

...and more generally,

### Exponents of exponents

You can also simplify an exponent of an exponent, this time you multiply the exponents. For example, 2 to the power of 3 all to the power of 4,

Again we can generalise to

### Different Bases

You **cannot** use the rules of multiplication and division with exponents which have different bases. For example,

The first exponent has the base 3 and the other 5, so you cannot simplify the expression.

However, if one of the bases is a power of the other, you can transform them into an expression where they have a common base. For example,

In this case, \(2^{3} = 8\)

Therefore you can replace the 8 in the original expression with \(2^{3}\)

And we know that we can simplify powers of exponents my multiplying them

Which gives us two exponents to multiply with the same base

And that's all you need to know about exponents for the GMAT.

In the multiplication of Exponent, Can you please give examples where Bases are different? When the bases are different - what rules apply ?

Sorry - Kindly ignore my previous question as I realized that you have provided examples & rules for different bases below. Sincere Apologies.

How will u simplify the first example given by you for the exponents when they have different bases.. i.e. (e.g. 3 raised to the power of 4 * 5 raised to the power of 2) ?

In general you can't simplify the expression (apart from just doing the calculation) when you have exponents with different bases.

I need help with this question:

If (1/5)^m(1/4)^18 = 1/(2(10)^35), then m = ?

Coreyeck, try this question which is a little easier but can be solved in exactly the same manner similar as your question above.

Read the explanation if you struggle with it and then you should be able to have a go at your question.

i don't understand how he gets 2^3 to equal 8,im completely lost because of that when it comes to the different bases

2 to the power of 3 is 2 times itself 3 times so

See common exponents for more details.

how to solve (1-(-7)^k+1+4*2*(-7)^k+1)/4