**Laws of Indices**

**Law 1**

$x^1 = x$

Any real number to the power of 1 is equal to itself e.g. $6^1 = 6$

** Law 2**

$x^0 = 1$

Any real number to the power of zero is always equal to 1 e.g. $7^0 = 1$

**Law 3**

$x^{-1} =\frac{1}{x}$

Any real number to the power of $-1$ is equal to the reciprocal of the base e.g. $4^{-1}=\frac{1}{4}$

**Law 4**

$x^mx^n = x^{m+n}$

If the bases are the same and the two number are multiplying each other, then add their respective powers e.g. $x^2x^3 = x^{2+3} = x^5$

**Law 5**

$\frac{x^m}{x^n} = x^{m-n}$

If the bases are the same and the two number are dividing each other, then subtract the power of the denominator from that of the numerator e.g. $\frac{x^6}{x^2} = x^{6-2} = x^4$

** Law 6**

$(x^m)^n = x^{mn}$

The power of a power implies the multiplication of the the two powers e.g. $(x^2)^3 = x^{2 \times 3} = x^6$

**Law 7**

$(xy)^n = x^ny^n$

A composite base inside brackets raised to a power implies that each factor in the base is raised to that power i.e. you can split the base with each of the factors raised to the same power e.g. $(xy)^3 = x^3y^3$

**Law 8**

$(\frac{x}{y})^n = \frac{x^n}{y^n}$

A fraction raised to a power implies that both the denominator and numerator are each raised to that power e.g. $(\frac{x}{y})^2 = \frac{x^2}{y^2} $

** Law 9**

$x^{-n} = \frac{1}{x^n}$

Any real number to the power of $-n$ is equal to the reciprocal of the base raised to the power $n$ e.g. $x^{-3} = \frac{1}{x^3}$

**Law 10**

$x^{\frac{1}{n}} = \sqrt[n]{x}$

Any number raised to a fractional power $\frac{1}{n}$ is equal to the nth root of the base e.g.

$8^{\frac{1}{3}} = \sqrt[3]{8}$

**Law 11**

$x^{\frac{m}{n}} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m$

Any number raised to a fractional power $\frac{m}{n}$ is equal to the nth root of the base raised to the power $m$ e.g. $4^{\frac{3}{2}} = \sqrt[2]{4^3} = (\sqrt[2]{4})^3$

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