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$$5^{3}=5\times 5\times 5$$
5 is called the base

3 is called the exponent, also known as the index, or power.

Any non-zero number raised to the power of zero yields 1. $$\left( base\neq 0\right) ^{0}=1$$

The numbers in a number system are ordered, that is, they go like, one, two, three and so on.

The cardinals are a set of numbers. The number of elements in a set (3, 7 etc.)

The numbers are of different types. Study the map below; you will find them very useful.

The letters R, Q, N, and Z refers to a set of numbers such that:

R = real numbers include all numbers from negative infinity to positive infinity

Q= rational numbers wriiten as fractions

N = Natural numbers are all positive integers starting from 1. (1,2,3....to infinity)

z = integers ( all integers positive and negative ( -infinity, ..., -2,-1,0,1,2....infinity)

A surd is an irrational number that has a radical the square root sign. Recall that irrational numbers are $\pi$ $\sqrt{2}$ $\sqrt [3] {5}$ and so on.

The following rules apply to surds.

For x > 0 and y > 0,

$$\left( \sqrt {x}\right) ^{2}=\sqrt {x^{2}}=x$$

$$\sqrt {xy}=\sqrt {x}\times \sqrt {y}$$

$$\sqrt {\dfrac {x}{y}}=\dfrac {\sqrt {x}}{\sqrt {y}}$$

**Example**

$$\sqrt {3}\times \sqrt {10}=\sqrt {10\times 3}=\sqrt {30}$$

**Addition & Subtraction of surds**

Surds can be added to or subtracted from other surds. That is, only irrational numbers with the radical sign.

$$3\sqrt {2}+5\sqrt {2}=\sqrt {2}\left( 3+5\right) =8\sqrt {2}$$

Surds can be multiplied using the rules shown above. Use rules of algebra where applicable.

**Example**

Expand the following: \begin{align} (\sqrt{5}+3)(\sqrt{5}-3) &\cssId{Step1}{= (\sqrt{5})^2-(3^2)}\\ &\cssId{Step2}{= 5 - 9}\\ &\cssId{Step3}{= -4}\\ \end{align}

**Rationalising the denominator**

In order to rationalise, if the denominator is a surd, multiply both the numerator and the denominator by the surd in the denominator.

**Example**

Rationalize the denominator \begin{align} (\dfrac {5\sqrt {3}}{\sqrt {7}}) &\cssId{St1}{= \dfrac {5\sqrt {3}}{\sqrt {7}}\times \dfrac {\sqrt {7}}{\sqrt {7}}}\\ &\cssId{St2}{= \dfrac {5\sqrt {3}\times \sqrt {7}}{7}}\\ &\cssId{St3}{= \dfrac {5}{7}\sqrt {21}}\\ \end{align}

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