$ \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.