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:
,
,
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 
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:
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Example 2
Multiply 
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:
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= 
= 
= 
= 
Example 3
Multiply 
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:
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Example 4
Multiply ![Rendered by QuickLaTeX.com \sqrt [3] {27} \cdot \sqrt [3] {2}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%2083%2021'%3E%3C/svg%3E)
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:
= ![Rendered by QuickLaTeX.com \sqrt [3] {27 \cdot 2}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%2064%2021'%3E%3C/svg%3E)
= ![Rendered by QuickLaTeX.com \sqrt [3] {54}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%2039%2021'%3E%3C/svg%3E)
= ![Rendered by QuickLaTeX.com \sqrt [3] {3 \times 3 \times 3 \times 2 }](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20137%2021'%3E%3C/svg%3E)
= ![Rendered by QuickLaTeX.com \sqrt [3] {3^3 \times 2}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%2073%2021'%3E%3C/svg%3E)
= ![Rendered by QuickLaTeX.com 3\sqrt [3] {2}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%2040%2021'%3E%3C/svg%3E)
Example 5
Multiply 
Solution
We will simply multiply the terms inside the radical symbol together. The radicands in this example are 4y and
. The product of 4y and
is equal to
. Hence, the final answer will be:
= 
The answer can be simplified further because
is equal to
.
= 
= 
Example 6
Multiply 
Solution
We will simply multiply the terms inside the radical symbol together. The radicands in this example are 8x and
. The product of 8x and
is equal to
. Hence, the final answer will be:
= 
The answer can be simplified further because
is equal to
.
= 
= 
Example 7
Multiply 
Solution
We will simply multiply the terms inside the radical symbol together. The radicands in this example are
and
. The product of
and
is equal to
. Hence, the final answer will be:
= 
The answer can be simplified further because
is equal to
.
= 
= 
Example 8
Multiply ![Rendered by QuickLaTeX.com 2\sqrt {4} \cdot 8\sqrt [3] {3}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%2093%2021'%3E%3C/svg%3E)
Solution
= ![Rendered by QuickLaTeX.com 2 \times 2 \cdot 8 \sqrt [3] {3}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20101%2021'%3E%3C/svg%3E)
= ![Rendered by QuickLaTeX.com 32 \sqrt [3] {3}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%2051%2021'%3E%3C/svg%3E)
Example 9
Multiply ![Rendered by QuickLaTeX.com 5\sqrt {16} \cdot 3\sqrt [3] {4}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20104%2021'%3E%3C/svg%3E)
Solution
= ![Rendered by QuickLaTeX.com 2 \times 2 \times 5 \times 3 \sqrt [3] {4}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20148%2021'%3E%3C/svg%3E)
= ![Rendered by QuickLaTeX.com 60 \sqrt [3] {4}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%2051%2021'%3E%3C/svg%3E)
Example 10
Multiply 
Solution
We will simply multiply the terms inside the radical symbol together. The radicands in this example are
and
. The product of
and
is equal to
. Hence, the final answer will be:
= 
The answer can be simplified further because
is equal to
.
= 
= 
Example 11
Multiply ![Rendered by QuickLaTeX.com \sqrt [3] {125} \cdot \sqrt [3] {2}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%2093%2021'%3E%3C/svg%3E)
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:
= ![Rendered by QuickLaTeX.com \sqrt [3] {125 \cdot 2}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%2075%2021'%3E%3C/svg%3E)
= ![Rendered by QuickLaTeX.com \sqrt [3] {250}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%2049%2021'%3E%3C/svg%3E)
= ![Rendered by QuickLaTeX.com \sqrt [3] {5 \times 5 \times 5 \times 2 }](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20137%2021'%3E%3C/svg%3E)
= ![Rendered by QuickLaTeX.com \sqrt [3] {5^3 \times 2}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%2073%2021'%3E%3C/svg%3E)
= ![Rendered by QuickLaTeX.com 5\sqrt [3] {2}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%2039%2021'%3E%3C/svg%3E)
Example 12
Multiply ![Rendered by QuickLaTeX.com \sqrt [3] {9} \cdot \sqrt [3] {1000}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20104%2021'%3E%3C/svg%3E)
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:
= ![Rendered by QuickLaTeX.com \sqrt [3] {9 \cdot 1000}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%2085%2021'%3E%3C/svg%3E)
= ![Rendered by QuickLaTeX.com \sqrt [3] {1000}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%2060%2021'%3E%3C/svg%3E)
= ![Rendered by QuickLaTeX.com \sqrt [3] {2 \times 5 \times 2 \times 5 \times 2 \times 5 \times 3 \times 3 }](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20282%2021'%3E%3C/svg%3E)
= ![Rendered by QuickLaTeX.com \sqrt [3] {2^3 \times 3^3 \times 5^3 \times 9}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%20164%2021'%3E%3C/svg%3E)
= ![Rendered by QuickLaTeX.com 30\sqrt [3] {9}](data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%2051%2021'%3E%3C/svg%3E)
Example 13
Multiply 
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:
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= 
Example 14
Multiply 
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:
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= 
Example 15
Multiply 
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:
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