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Radicals

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Radical is a symbol used to indicate the square root or nth root. It is a symbol placed over a number or expression known as radicand that indicates the root of the radicand. Radical means root which can be square root, cube root, etc., If there is no number, the root is a square root. A fraction may contain a radical but a fraction is not a radical. Radicals are also known as surds.
Radical is a function which finds a divider of the argument which upped on exponent gives argument. Radicals are the opposite functions of exponent.

A radical expression is an expression containing square root, where the radical ($\sqrt{}$) symbol is used to denote the square roots or nth roots while radicand is an expression inside the radical symbol.
Product of two radicals with the same index number is:

$\sqrt[a]{l}\sqrt[a]{m}$ = $\sqrt[a]{lm}$

Examples of radical expressions: $\frac{2+5\sqrt{2}}{5}$, $\sqrt{y}$ + 7, 9 $\pm$ $\sqrt[8]{7}$
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Rules For Radicals

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Given below are the important rules of radicals:

1) $\sqrt[n]{a.b}$= $\sqrt[n]{a} . \sqrt[n]{b}$

2) $b$ = $\sqrt[n]{a}$; if both $b$ $\geq$ 0 and $b^{n}$ = $a$

3) $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}$ = $\sqrt[n]{\frac{a}{b}}$

4) $\sqrt{\frac{a}{b}}$ =$\frac{\sqrt{a}}{\sqrt{b}}$

5) $\sqrt{a}+\sqrt{b}$ $\neq$ $\sqrt{a+b}$

6) $b^{\frac{a}{n}}$ = $\sqrt[n]{b^{a}}$ = $(\sqrt[n]{b})^{a}$

7) $\sqrt[n]{a^{m}}$= $a$ $^{\frac{m}{n}}$

8) If $a$ $\geqslant$ 0 then $\sqrt[n]{a^{n}}$ = $a$

Operations with Radicals

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If there exists any number 'a' along with a positive Integer n, such that the nth root of the number 'a' is any irrational number, then we say that $\sqrt[n]{a}$ is the radical. When we write $\sqrt[n]{a}$, we say n is the order of the radical and 'a' is the radicand. We must remember that every radical which we write is irrational number but all irrational Numbers are not the radicands.

Different operations on radical expressions can be performed. Some of them are as follows:

1) $\sqrt[n]{a}$ $\times$ $\sqrt[n]{b}$ = $\sqrt[n]{a \times b}$, which means if we have $\sqrt[3]{2}$ $\times$ $\sqrt[3]{5}$ then it can be written as

$\sqrt[3]{2}$ $\times$ $\sqrt[3]{5}$ = $\sqrt[3]{2 \times 5}$

= $\sqrt[3]{10}$

It represents that if the cube root of 2 and cube root of 5 are multiplied, then it will be equal to the cube root of the product of 2 and 5 i.e. cube root of 10.

2) $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}$ = $\sqrt[n]{\frac{a}{b}}$

Example: $\frac{\sqrt[3]{2}}{\sqrt[3]{5}}$ = $\sqrt[3]{\frac{2}{5}}$

It represents that if the cube root of 2 is divided by cube root of 5, then it will be equal to the cube root of the quotient of 2 and 5 i.e. cube root of

$\frac{2}{5}$.

3) Finding the conjugate radical is another operation with radical expression, which means two radicals which only differ by the sign of + or – are called conjugate radicals.

For example: conjugate radical of $\sqrt{2}$ + $\sqrt{3}$ is $\sqrt{2}$ - $\sqrt{3}$

4) While performing operations with radical expressions, we must always remember that only radicals with similar order can be multiplied or divided and only similar radicals can be added or subtracted. If we have the radicals which are not of same order then we need to perform the operation of multiplication or division on them, we should first reduce them to the same order and then perform the multiplication or division operation.

5) If the product of the two radicals is taken out we get a rational number as the product, then each of the radical is called the rationalizing factor of the other radical.

Suppose we have $\sqrt{63}$ $\times$ $\sqrt{7}$= $\sqrt{441}$, which is the Square root of 21 and we know that 21 is the rational number and not

the irrational number, although the two numbers $\sqrt{7}$ and $\sqrt{63}$ are irrational.

Negative Radicals

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Negative radical is an imaginary number defined as $i$ = $\sqrt{-1}$
If there is a negative sign under the radical then it is 'i' since the problem contains square root of -1.
When you come across questions as what is the square root of - 5 then the correct answer is there is no real solution. However it can also be taken as
For Example:
Consider $\sqrt{-49}$

$\sqrt{-49}$ = $\sqrt{-1}$ . $\sqrt{49}$ = 7 $i$

Radical Problems

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Example 1:Simplify : $\sqrt{500}$

Solution: As 500 is not a perfect square, however it has a factor which can be written in terms of square root.
We find the largest perfect squares that divides into 500, that is 100.
500 in terms of product is 100 $\times$ 5

By product rule, $\sqrt{500}$ = $\sqrt{100.5}$

= $\sqrt{100}$ . $\sqrt{5}$

= 10$\sqrt{5}$

Example 2: Solve : $\frac{\sqrt{25}}{\sqrt{100}}$

Solution :

$\frac{\sqrt{25}}{\sqrt{100}}$ = $\sqrt{\frac{25}{100}}$

= $\sqrt{\frac{1}{4}}$

=$\frac{\sqrt{1}}{\sqrt{4}}$

=$\frac{1}{2}$