Question

How do you simplify $\sqrt{12}-\sqrt{147}?$

Answer 1

Hint: We will simplify the two values under the radical sign separately. We know that we can always write a composite number as a product of two numbers. The two values under the radical sign in the given problem are composite. We will check if any of the factors of these values is perfect square. We will use the identity given by $\sqrt{ab}=\sqrt{a}\sqrt{b}.$

Complete step by step solution:
Let us consider the given problem.
We are asked to simplify $\sqrt{12}-\sqrt{147}.$
Let us consider the values whose difference we need to find.
First let us take the value $\sqrt{12}.$
We know that the number $12$ is a composite number. We can write it as a product of two numbers. We know that $12=3\times 4.$
Also, the number $4$ is a perfect square. So, we can write $\sqrt{12}=\sqrt{3\times 4}.$
Now, we will use the identity given by $\sqrt{ab}=\sqrt{a}\sqrt{b}.$
Using this identity, we can write it as $\sqrt{12}=\sqrt{3}\sqrt{4}.$
We know that $\sqrt{4}=2.$
So, we will get $\sqrt{12}=2\sqrt{3}.$
Similarly, we will evaluate the next value.
We know that $147$ is also a composite number. Also, it can be written as the product $147=3\times 49.$
We know that the number $49$ is also a perfect square.
We are familiar with the fact that the square root of $49$ is $7.$
That is, $\sqrt{49}=7.$
We can write the value as $\sqrt{149}=\sqrt{3\times 49}.$
Let us use the identity $\sqrt{ab}=\sqrt{a}\sqrt{b}.$
We will get $\sqrt{149}=\sqrt{3}\sqrt{49}.$
We can write this as $\sqrt{149}=7\sqrt{3}.$
Now, we know that the given expression will become $\sqrt{12}-\sqrt{147}=2\sqrt{3}-7\sqrt{3}.$
Let us take the common term out. We will get $\sqrt{12}-\sqrt{147}=\left( 2-7 \right)\sqrt{3}.$
We know that $2-7=-5.$
Hence the simplified form of the given expression is $\sqrt{12}-\sqrt{147}=-5\sqrt{3}.$

Note: A number is called composite if it can be written as the product of two numbers other than $1$ and itself. A number that cannot be written as a product of $1$ and itself is called a prime number. A composite can be written as the product of two or more prime numbers.