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,

\[ \begin{split} 5^2 \times 5^4 &= ( 5 \times 5 ) \times ( 5 \times 5 \times 5 \times 5 ) \\ &= 5^{2 + 4} \\ &= 5^6 \end{split} \]

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

We can generalise this to

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

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,

\[ \begin{split} 7^5 \div 7^3 &= \frac{7 \times 7 \times 7 \times 7 \times 7}{7 \times 7 \times 7} \\ &= 7^{5 - 3} \\ &=7^2 \end{split} \]

...and more generally,

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

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,

\[ \begin{split} (2^3)^4 &= 2^{3 \times 4} \\ &= 2^{12} \end{split} \]

Again we can generalise to

\[ \begin{split} (x^n){^m} &= x^{n \times m} \\ &= x^{nm} \end{split} \]

Different Bases

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

\[3^{4} \times 5^{2}\]

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,

\[2^{5} \times 8^{2}\]

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

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

\[2^{5} \times 8^{2} = 2^{5} \times (2^{3})^{2}\]

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

\[ \begin{split} 2^5 \times (2^3)^2 &= 2^5 \times 2^{3 \times 2} \\ &= 2^5 \times 2^6 \end{split} \]

Which gives us two exponents to multiply with the same base

\[ \begin{split} 2^5 \times 2^6 &= 2^{5 + 6} \\ &= 2^{11} \end{split} \]

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

Next page: Ratios

Comments (9):

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

    hrishikeshgmat on 14 Dec 2008 (permalink)
  2. Sorry - Kindly ignore my previous question as I realized that you have provided examples & rules for different bases below. Sincere Apologies.

    hrishikeshgmat on 14 Dec 2008 (permalink)
  3. 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) ?

    hrishikeshgmat on 15 Dec 2008 (permalink)
  4. In general you can't simplify the expression (apart from just doing the calculation) when you have exponents with different bases.

    joel on 16 Dec 2008 (permalink)
  5. I need help with this question:

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

    Coreyeck on 10 Nov 2009 (permalink)
  6. 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.

    joel on 11 Nov 2009 (permalink)
  7. 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

    slade178 on 7 Jun 2010 (permalink)
  8. 2 to the power of 3 is 2 times itself 3 times so

    See common exponents for more details.

    joel on 8 Jun 2010 (permalink)
  9. how to solve (1-(-7)^k+1+4*2*(-7)^k+1)/4

    norazimah on 31 Jan 2013 (permalink)

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