# Dividing with Exponents

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Dividing with exponents involves simplification of exponents using the rule of exponents. Most of problems in mathematics deal with exponents. Simplification of exponents leads to solution easily. The most important rule on dividing the exponent is that while dividing the exponent with the same base we have to subtract the powers.  The division of exponents is opposite to multiplication because we add the powers when we multiply the exponents with same base.
The following examples clearly illustrate the division of expression with exponents.

Example 1: Divide 7x^5 / x^-3

Solution: 7x^5 / x^-3

To simplify this use rule of exponents

Divide the exponent having like bases; subtract the exponent of variable x in the denominator from the exponent of variable x in the numerator

X^5 / x^-3 = x^5-(-3) = x^5+3 = x^8

And 7/1= 7

Therefore, 7x^5 / x^-3 = 7x^8

Example 2: Divide : 4x^3y^2z^-5 / 2x^7yz

Solution: Use law of exponent to divide the like terms 4/2 = 2

x^3 / x^7 = x^3-7 = x^-4

y^2 / y = y^2-1 = y^1=y    (note that y = y^1 )

Then z^-5 / z = z^-5-1 = z^ -6

Combine all the terms together, we get 2x^-4yz^-6

This is the required solution.

Example 3: Divide 15a^3b^4a^2 / 3a^-1b^2

Solution: Notice that, numerator has 2 exponents with same base a^3 and a^2

Multiply the exponent first and then divide by the denominator

So, a^3.a^2 / a^-1 = a^3+2 – (-1) = a^6

15/3 = 5

b^4 / b^2 = b^4-2 = b^2

15a^3b^4a^2 / 3a^-1b^2 = 5 a^6 b^2.