Table of Contents

Last modified on October 28th, 2024

Exponents and square roots are inversely related operations. While exponents involve raising a number to a power, square roots inverse this operation by finding the original base that, when squared, gives the result.

For example,

8 to the 2nd power is 8^{2} = 64, which means 8 is multiplied by itself 2 times. When exponents are used, the results grow very rapidly.

Now, to find the square root of 64, we need a number that, when squared (multiplied by itself), equals 64. That number is 8, so the square root of 64 is 8.

When converting an exponent into a square root, the base, say x, is raised to the fractional exponent, ${x^{\dfrac{1}{2}}}$, which represents the square root of that base. It can be expressed as

**${x^{\dfrac{1}{2}}=\sqrt{x}}$**

In general, raising a base to the power of ${\dfrac{1}{n}}$ corresponds to finding the n^{th} root of the base.

For example,

${9^{\dfrac{1}{2}}=\sqrt{9}=3}$

To express a square root as an exponent, the base under the radical is rewritten with a fractional exponent. It is expressed as

**${\sqrt{x}=x^{\dfrac{1}{2}}}$**

More generally, an n^{th} root is written using a fractional exponent: ${\sqrt[n] {x}=x^{\dfrac{1}{n}}}$

For example,

${\sqrt{9}=9^{\dfrac{1}{2}}}$

However, to simplify complex problems involving these conversions, we need to follow certain rules.

Here are the rules for simplifying expressions involving both square roots and exponents.

**Simplify: ${\sqrt{x^{4}}}$**

Solution:

Here, ${\sqrt{x^{4}}}$

Converting the square root into an exponent,

${x^{\dfrac{4}{2}}}$

= x^{2}

Thus, ${\sqrt{x^{4}}}$ = x^{2}

**Simplify: ${\sqrt{16x^{6}}}$**

Solution:

Here, ${\sqrt{16x^{6}}}$

Using the product rule for radicals,

${\sqrt{16}\times \sqrt{x^{6}}}$

= ${4x^{\dfrac{6}{2}}}$

= ${4x^{3}}$

Thus, ${\sqrt{16x^{6}}}$ = ${4x^{3}}$

**Solve: ${\dfrac{\sqrt{x^{5}}}{\sqrt{x^{2}}}}$**

Solution:

Here, ${\dfrac{\sqrt{x^{5}}}{\sqrt{x^{2}}}}$

Using the quotient rule for square roots,

${\sqrt{\dfrac{x^{5}}{x^{2}}}}$

= ${\sqrt{x^{5-2}}}$

= ${\sqrt{x^{3}}}$

= ${x^{\dfrac{3}{2}}}$

Thus, ${\dfrac{\sqrt{x^{5}}}{\sqrt{x^{2}}}}$ = ${x^{\dfrac{3}{2}}}$

**Simplify the expression: ${\left( \sqrt{x^{6}}\right) ^{2}}$**

Solution:

Here, ${\left( \sqrt{x^{6}}\right) ^{2}}$

Using the power of a root rule,

= ${\left( \left( x^{6}\right) ^{\dfrac{1}{2}}\right) ^{2}}$

= ${\left( x^{\dfrac{6}{2}}\right) ^{2}}$

= ${\left( x^{3}\right) ^{2}}$

= ${x^{3\times 2}}$

= ${x^{6}}$

Thus, ${\left( \sqrt{x^{6}}\right) ^{2}}$ = ${x^{6}}$

Last modified on October 28th, 2024