$ \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 **s****quare 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.