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Radical Expressions and Equations

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In algebra, radicals are used quite commonly. The term radical is extracted from the Latin language word "radix" which refers to the "root". We know that the root is said to be the source of something; likewise if we find square or cube of a number (i.e. $3^{2}$ = $9$ or $3^{3}$ = $27)$, then the result $(9$ or $27)$ grows from the number $(3)$ is called the radicand. The terms radical and radicand were first used in mid $16^{th}$ century in England in a a book "An Introduction to Algebra" written by John Pell. Let's go ahead and learn more details about radicals and radical expressions.

Definition

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A radical expression is defined as any expression containing a radical symbol ($\sqrt{}$). Many people call it a square root symbol, but not always this symbol calculates square root. It may also used to describe a cube root, a fourth root or ever a higher root. Other than the square root, a radical symbol denotes a number in superscript. It means :

$\sqrt{x}$ = Square root of $x$
$\sqrt[3]{x}$ = Cube root of $x$
$\sqrt[4]{x}$ = Fourth root of $x$
and so on.

Let us suppose that $x$ be a number and $n$ be a non-negative integer. The nth root of $x$ is obtained by raising $n{th}$ power to a number (say k) yielding $x$. This is written as :
$k^{n}$ = $x$
this implies
$\sqrt[n]{x}$ = $k$


Operations With Radical Expressions

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Various operations with radical expressions can be dealt using the following rules.

Assume that $m$ and $n$ be two positive integers.

Multiplication Operations

The product of two or more numbers with same $m^{th}$ root can be written as the $m^{th}$ root of product of those numbers.
$\sqrt[m]{p} . \sqrt[m]{q} = \sqrt[m]{pq}$

Division Operations

The division of two numbers with same $m^{th}$ root can be expressed as the $m^{th}$ root of division of those numbers.
$\frac{\sqrt[m]{p}}{\sqrt[m]{q}} = \sqrt[m]{\frac{p}{q}}$

Exponentiation

The exponentiation operation can be expressed in the following way.
$p^{\frac{m}{n}} = (p^{m})^\frac{1}{n} = \sqrt[n]{p^{m}}$

Simplifying Radical Expression

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A radical expression can be referred to acquire a simplified form if it satisfies the following conditions.

(i) No fractions should be present under the radical sign.
   
    i.e $3$ $\sqrt{\frac{7}{5}}$ = $\frac{3 \sqrt{7}}{\sqrt{5}}$

(ii) In radicand, there should not be any factor that is able to be written in terms of a power bigger than or equal to the index.

For example :


$\sqrt{16}$ can be written as $\sqrt{2^{4}}$. So, this is not a simplified form.

Therefore, $\sqrt{16}$ = $\sqrt{2^{4}}$ = $4$

(iii) Any radical should not be present in the denominator. If it is so, the numerator and denominator are to be multiplied by the conjugate of the denominator (or the same radical as per the requirement) in order to eliminate radical from the denominator. This process in called rationalization of the denominator.

For example :

$\frac{3}{\sqrt{2}}$ = $\frac{3 \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}}$ = $\frac{3 \sqrt{2}}{2}$

Examples

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The examples based on radical expressions are given below.
Example 1 : 

Determine the value of $\sqrt{108}$.

Solution :
 

   $\sqrt{108}$

= $\sqrt{3 \times 36}$

= $6 \sqrt{3}$
Example 2 : 

Solve $\sqrt[3]{8x^{3}y^{3}} \times \sqrt{16x^{5}}$.

Solution :
 

   $\sqrt[3]{8x^{3}y^{3}} \times \sqrt{16x^{5}}$

= $\sqrt[3]{2^{3}x^{3}y^{3}} \times \sqrt{4^{2}x^{4}x}$

= $2xy \times 4 x^{2}\sqrt{x}$

= $8x^{3}y \sqrt{x}$
Example 3 : 

Simplify the radical expression $\frac{5}{3+ \sqrt{7}}$.

Solution :
 

$\frac{5}{3+ \sqrt{7}}$

Multiplying and dividing the numerator and denominator by $3$ - $\sqrt{7}$ which the conjugate of $3$ + $\sqrt{7}$.

$\frac{5(3- \sqrt{7})}{(3+ \sqrt{7})(3- \sqrt{7})}$

Applying the formula $(a + b)(a - b)$ = $a^{2}$ - $b^{2}$

= $\frac{15- 3 \sqrt{7}}{3^{2} - (\sqrt{7})^{2}}$

= $\frac{15- 3 \sqrt{7}}{9- 7}$

= $\frac{15- 3 \sqrt{7}}{2}$