# Power of a Quotient Propertyand 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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