In this article, you will learn what are radicals, how to add, subtract, multiply and divide them, and how to simplify radical expressions containing powers. But before proceeding to discuss that, first, we will see what are radical expressions.

What are Radicals?

Radical expressions are inverse of exponents and the result of these expressions are roots. In radical expressions, the numbers or coefficients are written inside the radical sign \sqrt. Radicals are also known as roots. We can convert radical expressions to exponents and exponents into radicals.

For example, the square of 9 is 81 and the square root of 81 is 9. Similarly, if we take the square of 5x, we will get 25x^2 and if we will take the square root of 25x^2, then we will get 5x.

We can read the expression \sqrt{25} as root 25, square root of nine or radical nine. We can also take the powers of the numbers other than two, for instance, 3, 4, 5, 6 and so on.

How to add, subtract, multiply and divide radicals?

We often come across the fractions that contain radicals in the numerator and the denominator. When the radicals are present in the denominator, then it becomes much easier to simplify the expression by rewriting it in a way that the radical sign is eliminated from the denominator.

Simplifying Radical Expressions

We can simplify a radical expression if:

  • The radicand, i.e. the value inside the radical sign can be converted into an exponent
  • The radicand is a fraction
  • The denominator of a fraction has a radical sign

Let us solve some examples in which we will add, subtract, multiply and divide the radical expressions. In these example, we will also see how to simplify the radical expressions containing powers.

Example 1

Add the following radical expression:

8 \sqrt{6} + 9 \sqrt {6}

Solution

To add the above expressions, first we will see whether the radicand, i.e. the number inside the radical sign is same or not. In the above case, this number is same, i.e. 6. Hence, we will simply add the number 9 and 8 and keep \sqrt{6} as it is:

8 \sqrt{6} + 9 \sqrt {6} = 17 \sqrt {6}

 

 

Example 2

Simplify the following radical expression:

2 \sqrt{5} - 3 \sqrt {3} - 7 \sqrt {3}

Solution

To subtract the above expressions, first we will see whether the radicands, i.e. the number inside the radical sign is same or not. In the above case, this number is same, i.e. 3 for two terms and it is different for one term. Hence, we will simply evaluate the expressions -3 \sqrt {3} and - 7 \sqrt{3} and keep the other term with different radicand 2 \sqrt {5} as it is:

= 2 \sqrt{5} - 3 \sqrt {3} - 7 \sqrt {3}

= 2 \sqrt {5} - 10 \sqrt {3}

 

Example 3

Simplify the following radical expression:

7 \sqrt{8} + 2\sqrt {8} - 6 \sqrt {4}

Solution

To add or subtract the above expressions, first, we will see whether the radicands, i.e. the numbers inside the radical sign are same or not. In the above case, the number 8 is same for two terms and it is different for one term. Hence, we will simply evaluate the expressions 7 \sqrt{8} + 2\sqrt {8} together and simplify 6\sqrt {4} separately:

= 7 \sqrt{8} + 2\sqrt {8} - 6 \sqrt {4}

= 9 \sqrt {8} - 6 \sqrt {4}

\sqrt {4} is equal to 2:

= 9  \sqrt {8} - 6 \cdot 2

= 9 \sqrt {8} - 12

\sqrt {8} is equal to \sqrt {2^2 \cdot 2}

= 9 \sqrt {2^2 \cdot 2} - 12

= 18 \sqrt {2} - 12

 

Example 4

Simplify the following radical expression:

3 \sqrt {7} \cdot 9 \sqrt {2}

Solution

To multiply the above expression, we will multiply the terms outside the radical sign together. We will repeat the same for the terms inside the radical sign. The resulting answer will be:

= 3 \sqrt {7} \cdot 9 \sqrt {2}

= 27 \sqrt {14}

 

Example 5

Simplify the following radical expression:

\frac {\sqrt {5}} {\sqrt{8}}

Solution

In this expression, we have got a radical sign in the denominator. Hence, we will use the process of rationalization to simplify this expression. First, we will multiply and divide the expression by the term in the denominator like this:

= \frac {\sqrt {5}} {\sqrt{8}}

= \frac {\sqrt {5}} {\sqrt{8}} \cdot \frac {\sqrt {8}} {\sqrt{8}}

= \frac {\sqrt {40}} {8}

\sqrt {40} is equal to \sqrt {2^2 \cdot 10} = 2 \sqrt {10}.

= \frac {2\sqrt {10}} {8}

By simplifying it further, we will get the following answer:

= \frac {\sqrt {10}} {4}

 

Example 6

Simplify the following radical expression:

\frac {\sqrt {2}} {\sqrt{5}}

Solution

In this expression, we have got a radical sign in the denominator. Hence, we will use the process of rationalization to simplify this expression. First, we will multiply and divide the expression by the term in the denominator like this:

= \frac {\sqrt {2}} {\sqrt{5}}

= \frac {\sqrt {2}} {\sqrt{5}} \cdot \frac {\sqrt {5}} {\sqrt{5}}

= \frac {\sqrt {10}} {5}

 

Example 7

Simplify the following radical expression:

(\frac {\sqrt [3]{20} \cdot \sqrt [4] {27}} {\sqrt{8}})^4

Solution

In the first step, we will apply the power 4 to all the expressions separately like this:

= \frac {(\sqrt [3] {(20) ^4} \cdot \sqrt [4] {(27)^4}} {\sqrt {(8)^4}}

(\sqrt [3] {20} \sqrt [4] {27})^4 is equal to 540 \sqrt [3]{20}

= \frac {540 \sqrt [3]{20}} {(\sqrt {8})^4}

\sqrt {8}^4 is equal to 8^2:

= \frac {540 \sqrt [3]{20}} {8^2}

540 is equal to 3^3 \cdot 2^2 \cdot 5

= \frac {3^3 \cdot 2^2 \cdot 5 \sqrt [3]{20}} {8^2}

8^2 is equal to 2^6

= \frac {3^3 \cdot 2^2 \cdot 5 \sqrt [3]{20}} {2^6}

Simplify \frac {2^2}{2^6} like this:

= \frac {3^3 \cdot 5 \sqrt [3]{20}} {2^4}

3^3 \cdot 5^3 \sqrt {20} is equal to 135 \sqrt[3] {20}:

= \frac {135 \sqrt [3] {20}}{2^4}

2^4 is equal to 16.

= \frac {135 \sqrt [3] {20}}{16}

 

Example 8

Simplify the following radical expression:

(\frac{\sqrt {25} \cdot \sqrt [3] {8}} {\sqrt {10}})^3

Solution

\sqrt {25} is equal to 5 and \sqrt [3] {8} is equal to 2:

= (\frac {10} {\sqrt {10}})^3

Take the cube of the numerator and the denominator separately like this:

= \frac {(10)^3} {(\sqrt {10})^3}

= \frac {1000} {\sqrt {10}^3}

(\sqrt {10})^3 is equal to 10 \sqrt {10}

= \frac {1000} {10 \sqrt {10}}

= \frac {100} {\sqrt {10}}

100 is equal to 2^2 \cdot 5^2:

= \frac {2^2 \cdot 5^2} {\sqrt {2 \cdot 5}}

= \frac {2^2 \cdot 5^2} {\sqrt {2} \cdot \sqrt {5}}

\frac {2^2}{\sqrt{2}} is equal to 2 ^ {\frac {2}{3}} and \frac {5^2}{\sqrt{5}} is equal to 5 ^ {\frac {2}{3}}.

= 2 ^ {\frac {3}{2}} \cdot 5 ^ {\frac {3}{2}}

2 ^ {\frac {3}{2}} is equal to 2 \sqrt {2} and 5^ {\frac {3}{2}} is equal to 5 \sqrt {5}:

= 2 \sqrt {2} \cdot 5 \sqrt {5}

= 10 \sqrt {10}

 

Example 9

Simplify the following radical expression:

(\frac {10 \sqrt {12}} {\sqrt [3] {27}})^3

Solution

Apply the exponent 3 separately to the numerator and the denominator like this:

= (\frac {10 \sqrt {12}})^3 {(\sqrt [3] {27})^3}

10 (\sqrt {12})^3 is equal to 24000 \sqrt{3}

= \frac {24000 \sqrt {3}} {(\sqrt [3] {27})^3}

\sqrt [3] {27} = 27

= \frac {24000 \sqrt {3}} {27}

Cancel the common factor 3:

= \frac {8000\sqrt {3}} {9}

= \frac {8000} {3 ^ {\frac {3}{2}}}

3 ^ {\frac {3}{2}} is equal to 3\sqrt {3}:

= \frac {8000} {3\sqrt{3}}

 

Example 10

Simplify the following expression:

(\frac {5 \sqrt {4}} {\sqrt {10}})^3

Solution

(5 \sqrt {4})^3 is equal to 1000:

= \frac {1000} {(\sqrt {10})^3}

(\sqrt {10})^3 is equal to 10 \sqrt {10}

= \frac {1000} {10 \sqrt {10}}

= \frac {100}{\sqrt {10}}

100 is equal to 2^2 \cdot 5^2:

= \frac {2^2 \cdot 5^2} {\sqrt {2} \sqrt {5}}

\frac {2^2}{\sqrt{2}} is equal to 2^ {\frac {3}{2}} and \frac {5^2}{\sqrt{5}} is equal to 5^ {\frac {3}{2}}:

= 2^ {\frac {2}{3}} \cdot 5^ {\frac {2}{3}}

2^ {\frac {3}{2}} is equal to 2\sqrt {2} and 5^ {\frac {3}{2}} is equal to 5\sqrt {5}:

= 2\sqrt {2} \cdot 5 \sqrt {5}

Simplify the above expression by multiplying the radicands and terms outside the radical sign like this:

= 10 \sqrt {10}

 

 

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Rafia Shabbir