# Find the product of the following equation. Express it in standard form. Give the value of a followed by the value of b separated by a comma.

$\sqrt {30}\: and \: 6\sqrt {10}$

This article discusses the product of two numbers under the square root. The background concept used in this article is a simple product and square root method.

The product of $\sqrt {30}$ and $6 \sqrt {10}$ is $60 \sqrt {3}$.

The root product of a number is done by factoring the number so that the product of two identical numbers inside the root can be written as a single number.

The mathematical expression for the product of two equal numbers inside the root looks like this:

$\sqrt { a } . \sqrt { a } = ( \sqrt { a } ) ^ { 2 }$

$= a$

Similarly, the product of two numbers $\sqrt { 30 }$ and $6 \sqrt { 10 }$ can also be taken by factoring the number correctly.

Factorize the number $\sqrt { 30 }$ to its simplest form.

$\sqrt { 30 } = \sqrt { 3 \times 10 }$

$= \sqrt { 3 } . \sqrt { 10 }$

These two numbers can now be multiplied as shown below:

$\sqrt { 30 } \times \ 6 \sqrt { 10 } = \sqrt { 3 } . \sqrt { 10 } \times 6 \sqrt { 10 }$

$= \sqrt { 3 } \times ( 10 \times 6 )$

$= 60 \sqrt { 3 }$

Compare the value of the product to the standard form $a \sqrt { b }$.

$a \sqrt { b } = 60 \sqrt { 3 }$

$a=60 , b=2$

Thus, the product of $\sqrt { 30 }$ and  $6 \sqrt { 10 }$ in standard form is $60 \sqrt { 3 }$ and the value $a$  and $b$ are $60$ and $3$, respectively.

## Numerical Result

The product of $\sqrt{30}$ and $6\sqrt { 10 }$ in standard form is $60 \sqrt { 3 }$ and the value $a$  and $b$ are $60$ and $3$, respectively.

## Example

Find a product of $\sqrt { 20 }$ and $10\sqrt {5}$. Express it in standard form. Enter the a value followed by the b value, separated by a comma.

Solution

The product of $\sqrt 20$ and $10\sqrt 5$ is $50\sqrt 4$.

Factorize the number $\sqrt { 20 }$ to its simplest form.

$\sqrt { 20 } = \sqrt { 4\times 5 }$

$= \sqrt { 4 } . \sqrt { 5 }$

These two numbers can now be multiplied as shown below:

$\sqrt { 20 } \times 10\sqrt {5}=\sqrt{4}.\sqrt{5}\times 10\sqrt{5}$

$= \sqrt { 4 } \times ( 10 \times 5 )$

$= 50\sqrt {4}$

Compare the value of the product to the standard form $a\sqrt {b}$.

$a\sqrt {b}=50\sqrt {4}$

$a=50,b=4$

Thus, the product of $\sqrt {20}$ and $10\sqrt {5}$ in standard form is $50\sqrt {4}$ and the value $a$ and $b$ are $50$ and $4$, respectively.

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