Question

How do you simplify \[7\sqrt{28}\]?

Answer 1

Hint: Assume the given expression as ‘E’. Now, leave the number 7 as it is and find the square root of 28. To find the square root use the prime factorization method to write the number 28 as the product of its prime factors. Try to group the identical factors so that it can be written in the exponential form having exponent 2. Leave the factor which cannot be grouped and use the formula of exponents given as: - \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}\] to simplify the grouped factors.

Complete step by step solution:
Here, we have been provided with the expression \[7\sqrt{28}\] and we are asked to simplify it. Now, let us assume this expression as ‘E’, so we have,
\[\Rightarrow E=7\sqrt{28}\]
As we can see that here we have to find the square root of 28 to simplify the expression. So, let us use the prime factorization method to find the square root.
Now, in mathematics prime factorization is the method of writing a composite number as the product of its prime factors. So, we can write 28 as: -
\[\Rightarrow 28=2\times 2\times 7\]
Clearly, we can see that 2 and 7 are prime numbers. Now, we need to find the square root of 28, so we have to make a group of two identical factors. Therefore, we have,
\[\Rightarrow 28={{2}^{2}}\times 7\]
Here, 7 cannot be grouped so we will leave it as it is. Therefore, the expression ‘E’ becomes,
\[\Rightarrow E=7\sqrt{{{2}^{2}}\times 7}\]
In exponential form we can write the above expression as: -
\[\Rightarrow E=7\times {{\left( {{2}^{2}} \right)}^{\dfrac{1}{2}}}\times {{7}^{\dfrac{1}{2}}}\]
Using the formula of exponent: - \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}\], we get,
\[\begin{align}
  & \Rightarrow E=7\times {{2}^{2\times \dfrac{1}{2}}}\times \sqrt{7} \\
 & \Rightarrow E=7\times 2\times \sqrt{7} \\
 & \Rightarrow E=14\sqrt{7} \\
\end{align}\]
Hence, the above relation is the simplified form and our answer.

Note: One may note that here we have tried to form a group of two identical factors because we were to find the square root. If the expression would have contained cube root then we would have tried to make a group of three identical factors. You must remember the process of finding the prime factors of a number because here the number under the radical sign is small but sometimes it will be large numbers containing 4 – 5 digits and at those places prime factorization helps.