Law of Exponent Tips and Tricks

This topic is about how we exploit laws of exponent.Yes! This topic is sometimes confusing especially when it comes to its many rules. When to add, subtract, multiply, and divide exponents? At the end of this post I hope that those confusions will be cleared out.

Laws of Exponent

I. a^m\cdot a^n=a^{m+n}

We can only add exponents if the bases are the same.

Example: 3^x(3^{x^2})(3)=3^{x+x^2+3}


II. \displaystyle\frac{a^m}{a^n}=a^{m-n}

We can only subtract exponents if the bases are the same.

Example: \displaystyle\frac{a^{5x}}{a^{3y}}=a^{5x-3y}


III. (a^m)^n=a^{mn}

The base now doesn’t matter for us to multiple the exponents. If the exponent is enclosed by another exponent  and that is the time we divide them.

Example: (3^{2x})^2=3^{4x}


IV. (a^m)^\frac{1}{n}=a^\frac{m}{n}

We divide the exponents if an exponent is enclosed by another exponent less than greater than 0 and less than 1 or fractional exponent.

Example: \sqrt{4x^2y^4}=(4x^2y^2)^{\frac{1}{2}}=4^{\frac{1}{2}}(x^2)^ {\frac{1}{2}}(y^2)^ {\frac{1}{2}}=2xy


VI. a^-m=\displaystyle\frac{1}{a^m}

If the exponent is negative, we expressed the term to its reciprocal and change the sign to positive exponent.

Example: 3x^-2=\displaystyle\frac{3}{x^2}



If you see negative exponent, to make it positive change its orientation to make it positive. If the term with negative exponent is in the numerator transfer it to the denominator and change the sign of exponent to positive. This is my technique to keep my accuracy when dealing with negative exponents.


Worked Problem 1:

Simplify to lowest term and express the answer in positive exponents.




Make all exponents inside the parenthesis positive by changing the place of the terms.


Apply the laws of exponent

(\displaystyle\frac{x^{(2-4)}y^{(5-2)}}{12})^{-2} (\displaystyle\frac{x^{-2}y^{-3}}{12})^{-2}

\displaystyle\frac{x^{-2(-2)}y^{-3(-2)}}{12^{-2}} by distributing the exponent further evaluation.


Worked Problem 2:

Simplify: \displaystyle\frac{3^x+3^{x-1}}{3^{x+1}}


\displaystyle\frac{3^x(1+3^{-1})}{3^x\cdot 3} by factoring the common 3^x and cancel it out





Worked Problem 3:

Simplify: \sqrt{\displaystyle\frac{24x^2y^8}{6x^{-4}y^{-6}}}

We always start inside making the sign positive. The negative exponent will go to the numerator.

\sqrt{4x^{2+4}y^{8+6}} \sqrt{4x^6y^14} (4x^6y^14)^{\frac{1}{2}} 2x^3y^7










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