Chapters

What are Radicals?

The radical is also known as a root and it is the inverse operation of applying exponent. It means that we can remove power by taking a radicals and we can remove a radical by taking power. For example, if we take a square of 4, we will get 16 and when we will take a square root of 16, we will get 4. Similarly, if we take the cube of 2, we will get 8 and if we will get a cube root of 8, we will get 2. Mathematically, we can write these examples like this:

(4)^2 = 16 , \sqrt{16} = 4

(2)^3 = 8, \sqrt [3] {8} = 2

\sqrt{} is the radical symbol and the number inside this symbol is known as the radicand. We can read the expression \sqrt{16} as square root 16, root 16, or radical 16.
In the next section, we will discuss how to multiply radical expressions.
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Multiplying Radicals

We can apply the operations of addition, subtraction, multiplication, and division on radicals just like numbers. Radicals are multiplied together by multiplying their radicands (the terms inside the radical symbol). The product of radicands is kept under the same radical sign. Now, let us see how to multiply radicals practically through a couple of examples.

Example 1

Multiply \sqrt {6} \cdot \sqrt {10}

Solution

We will simply multiply the terms inside the radical symbol together. The radicands in this example are 6 and 10. The product of 6 and10 is equal to 60. Hence, the final answer will be:
= \sqrt {6 \cdot 10}
= \sqrt{60}
\sqrt{60} can be simplified further because 60 is equal to 2 x 3 x 5 x 2.
= \sqrt {2 \times 2 \times 3 \times 5}
= \sqrt {2^2 \times 15}
= 2\sqrt {15}

Example 2

Multiply \sqrt {8} \cdot \sqrt {6}

Solution

We will simply multiply the terms inside the radical symbol together. The radicands in this example are 8 and 6. The product of 8 and 6 is equal to 48. Hence, the final answer will be:
= \sqrt {8 \cdot 6}
= \sqrt{48}
\sqrt{48} can be simplified further because 48 is equal to 2 x 2 x 2 x 2 x 3.
= \sqrt {2 \times 2 \times 2 \times 2 \times 3}
= \sqrt {2^2 \times 2^2 \times 3}
= 4\sqrt {3}

Example 3

Multiply \sqrt {9} \cdot \sqrt {5}

Solution

We will simply multiply the terms inside the radical symbol together. The radicands in this example are 9 and 5. The product of 9 and 5 is equal to 45. Hence, the final answer will be:
= \sqrt {9 \cdot 5}
= \sqrt{45}
\sqrt{45} can be simplified further because 45 is equal to 3 x 3 x 5.
= \sqrt {3 \times 3 \times 5 }
= \sqrt {3^2 \times 5}
= 3\sqrt {5}

Example 4

Multiply \sqrt [3] {27} \cdot \sqrt [3] {2}

Solution

We will simply multiply the terms inside the radical symbol together. The radicands in this example are 27 and 2. The product of 27 and 2 is equal to 54. Hence, the final answer will be:
= \sqrt [3] {27 \cdot 2}
= \sqrt [3] {54}
\sqrt{54} can be simplified further because 54 is equal to 3 x 3 x 3 x 2.
= \sqrt [3] {3 \times 3 \times 3 \times 2 }
= \sqrt [3] {3^3 \times 2}
= 3\sqrt [3] {2}

Example 5

Multiply \sqrt {4y} \cdot \sqrt {6y^3}

Solution

We will simply multiply the terms inside the radical symbol together. The radicands in this example are 4y and 6y^3. The product of 4y and 6y^3 is equal to 24y^4. Hence, the final answer will be:
= \sqrt {24 y^4}
The answer can be simplified further because 24y^4 is equal to 2 \times 2 \times 2 \times 3 \times y^2 \times y^2.
= \sqrt {2 \times 2 \times 2 \times 3 \times y^2 \times y^2}
= 2y^2\sqrt {6}

Example 6

Multiply \sqrt {8x} \cdot \sqrt {5x^3}

Solution

We will simply multiply the terms inside the radical symbol together. The radicands in this example are 8x and 5x^3. The product of 8x and 5x^3 is equal to 40x^4. Hence, the final answer will be:
= \sqrt {40 x^4}
The answer can be simplified further because 40x^4 is equal to 2 \times 2 \times 2 \times 5 \times x^2 \times x^2.
= \sqrt {2 \times 2 \times 2 \times 5 \times x^2 \times x^2}
= 2x^2 \sqrt {10}

Example 7

Multiply \sqrt {2x^2} \cdot \sqrt {10x^3}

Solution

We will simply multiply the terms inside the radical symbol together. The radicands in this example are 2x^2and 10x^3. The product of 2x^2 and 10x^3 is equal to 20x^5. Hence, the final answer will be:
= \sqrt {20 x^5}
The answer can be simplified further because 20 x^5 is equal to 2 \times 2 \times 5 \times x^2 \times x \times x^2.
= \sqrt {2 \times 2 \times 5 \times x^2 \times x \times x^2}
= 2x^2 \sqrt {5x}

Example 8

Multiply 2\sqrt {4} \cdot 8\sqrt [3] {3}

Solution

\sqrt {4} is equal to 2. Hence, 2 multiplied by \sqrt{4} is equal to 4. It can be written as:
= 2 \times 2 \cdot 8 \sqrt [3] {3}
= 32 \sqrt [3] {3}

Example 9

Multiply 5\sqrt {16} \cdot 3\sqrt [3] {4}

Solution

\sqrt {16} is equal to 4. Hence, 5 multiplied by \sqrt{16} is equal to 20. It can be written as:
= 2 \times 2 \times 5 \times 3 \sqrt [3] {4}
= 60 \sqrt [3] {4}

Example 10

Multiply \sqrt {7x^2} \cdot \sqrt {10x^3}

Solution

We will simply multiply the terms inside the radical symbol together. The radicands in this example are 7x^2and 10x^3. The product of 7x^2 and 10x^3 is equal to 70x^5. Hence, the final answer will be:
= \sqrt {70 x^5}
The answer can be simplified further because 70 x^5 is equal to 2 \times 5 \times 7 \times x^2 \times x \times x^2.
= \sqrt {2 \times 5 \times 7 \times x^2 \times x \times x^2}
= x^2\sqrt {70x}

Example 11

Multiply \sqrt [3] {125} \cdot \sqrt [3] {2}

Solution

We will simply multiply the terms inside the radical symbol together. The radicands in this example are 125 and 2. The product of 125 and 2 is equal to 250. Hence, the final answer will be:
= \sqrt [3] {125 \cdot 2}
= \sqrt [3] {250}
\sqrt [3] {250} can be simplified further because 250 is equal to 5 x 5 x 5 x 2.
= \sqrt [3] {5 \times 5 \times 5 \times 2 }
= \sqrt [3] {5^3 \times 2}
= 5\sqrt [3] {2}

Example 12

Multiply \sqrt [3] {9} \cdot \sqrt [3] {1000}

Solution

We will simply multiply the terms inside the radical symbol together. The radicands in this example are 9 and 1000. The product of 9 and 1000 is equal to 9000. Hence, the final answer will be:
= \sqrt [3] {9 \cdot 1000}
= \sqrt [3] {1000}
\sqrt [3] {9000} can be simplified further because 9000 is equal to 2 x 5 x 2 x 5 x 2 x 5 x 3 x 3.
= \sqrt [3] {2 \times 5 \times 2 \times 5 \times 2 \times 5 \times 3 \times 3 }
= \sqrt [3] {2^3 \times 3^3 \times 5^3 \times 9}
= 30\sqrt [3] {9}

Example 13

Multiply \sqrt {8} \cdot \sqrt {9}

Solution

We will simply multiply the terms inside the radical symbol together. The radicands in this example are 8 and 9. The product of 8 and 9 is equal to 72. Hence, the final answer will be:
= \sqrt {8 \cdot 9}
= \sqrt{72}
\sqrt{72} can be simplified further because 72 is equal to 2 x 2 x 2 x 3 x 3.
= \sqrt {2 \times 2 \times 2 \times 3 \times 3}
= \sqrt {2 \times 3^2 \times 2^2}
= 6\sqrt {2}

Example 14

Multiply \sqrt {7} \cdot \sqrt {8}

Solution

We will simply multiply the terms inside the radical symbol together. The radicands in this example are 7 and 8. The product of 7 and 8 is equal to 56. Hence, the final answer will be:
= \sqrt {7 \cdot 8}
= \sqrt{56}
\sqrt{56} can be simplified further because 56 is equal to 7 x 2 x 2 x 2.
= \sqrt {7 \times 2 \times 2 \times 2 }
= \sqrt {2 \times 2^2 \times 7}
= 2\sqrt {14}

Example 15

Multiply \sqrt {8} \cdot \sqrt {10}

Solution

We will simply multiply the terms inside the radical symbol together. The radicands in this example are 8 and 10. The product of 8 and 10 is equal to 80. Hence, the final answer will be:
= \sqrt {8 \cdot 10}
= \sqrt{80}
\sqrt{80} can be simplified further because 80 is equal to 2 x 2 x 2 x 5 x 2.
= \sqrt {2 \times 2 \times 2 \times 2 \times 5 }
= \sqrt {2^2 \times 2^2 \times 5}
= 4\sqrt {5}
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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.