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Power of a Quotient Property
and other Power Properties


With powers in Math, there are some useful properties to know of that can make performing sums a bit quicker and easier much of the time.

Arguably the most useful of the properties to remember on this page will be the ‘powers of a quotient property’ and the ‘quotient of powers property’.
They’ll be referenced towards the end of the page, but first we’ll look at some other handy properties and rules.



Multiplying with same Base Value:

When terms with powers have the same base value, the powers are summed up together.

    General Rule:     a^m \times a^n  =  a^{m+n}

22 × 23   =   ( 2 × 2 ) × ( 2 × 2 × 2 )   =   4 × 8   =   32

22 × 23   =   ( 2 × 2 ) × ( 2 × 2 × 2 )  =   2 × 2 × 2 × 2 × 2   =   25   =   32




Power of a Power Property:

When terms with powers have the same base value, the powers are summed up together.

    General Rule:     (a^m)^n  =  a^{m \times n}

(22)3   =   ( 2 × 2 ) × ( 2 × 2 × 2 )   =   4 × 8   =   32

22 × 23   =   ( 2 × 2 ) × ( 2 × 2 × 2 )  =   2 × 2 × 2 × 2 × 2   =   25   =   32




Power of a Product Property:

When a product is involved with a power applied, the power applies to each part of the product individually.


(4x)^2   =   4^2x^2   =   16x^2

(4xy)^2   =   4^2x^2y^2   =   16x^2y^2

(3x^2yz^3)^2   =   3^2(x^2)^2y^2(z^3)^2   =   9x^4y^2z^6







Quotient Properties



Quotient of Powers Property:

If terms involving a power in a quotient have the same base value/number, then we can subtract the lower power from the upper one.

    General Rule:     \frac{a^{\tiny{m}}}{a^{\tiny{n}}}   =   a^{m \space {\text{--}} n}


\bf{\frac{2^{\tiny{4}}}{2^{\tiny{2}}}}  =  \bf{\frac{16}{4}}  =  4

\bf{\frac{2^{\tiny{4}}}{2^{\tiny{2}}}}  =  \bf{2^{4 \space {\text{--}} \space 2}}  =  \bf{2^2}  =  4

\frac{x^{\tiny{5}}y^{\tiny{6}}}{x^{\tiny{2}}y^{\tiny{4}}}   =   x^{5 \space {\text{--}} 2}y^{6 \space {\text{--}} 4}   =   x^{3}y^{2}




Power of a Quotient Property:

When a power is applied to a quotient inside brackets, the power applies to both the upper and lower values.

    General Rule:     (\frac{a}{b})^n   =   \frac{a^{\tiny{n}}}{b^{\tiny{n}}}


(\frac{1}{2})^{\tiny{2}}   =   \frac{1^{\tiny{2}}}{2^{\tiny{2}}}  =  \frac{1}{4}

(\frac{a^{\tiny{4}}}{a^{\tiny{3}}})^{\tiny{3}}   =   \frac{(a^{\tiny{4}})^{\tiny{3}}}{(a^{\tiny{3}})^{\tiny{3}}}  =  \frac{a^{\tiny{12}}}{a^{\tiny{9}}}  =  a^{12 \space {\text{--}} \space 9}  =  a^{3}





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