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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.

Expert Answer

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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