Question

How do you simplify $\sqrt{1440}$?

Answer 1

Hint: First we will assume the given expression as E. Now, we will write 1440 as the product of its primes by using the prime factorization process. We will try to form a group of two similar prime factors and use the formula ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$ to simplify. The factors whose group we may not be able to form will be left inside the radical sign.

Complete step-by-step answer:
Here, we have been provided with the expression $\sqrt{1440}$ and we are asked to simplify it.
Let us assume this expression as E. So, we have,
$\Rightarrow E=\sqrt{1440}$
Since, we need to find the square root of 1440 so we will apply the prime factorization method to do this. In this method first we will write 1440 as the product of its primes and try to form a group of two similar factors that will appear. So, we have,
$\Rightarrow 1440=2\times 2\times 2\times 2\times 2\times 3\times 3\times 5$
Forming the group of two similar prime factors and writing them in the exponential form with exponent equal to 2, we get,
\[\Rightarrow 1440={{2}^{2}}\times {{2}^{2}}\times 2\times {{3}^{2}}\times 5\]
Here, we are not able to group 2 and 5 so they will be left as it is. Therefore, the expression ‘E’ becomes:
\[\begin{align}
  & \Rightarrow E=\sqrt{{{2}^{2}}\times {{2}^{2}}\times 2\times {{3}^{2}}\times 5} \\
 & \Rightarrow E={{\left( {{2}^{2}}\times {{2}^{2}}\times 2\times {{3}^{2}}\times 5 \right)}^{\dfrac{1}{2}}} \\
\end{align}\]
Using the property of exponents given as ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}$, we get,
\[\begin{align}
  & \Rightarrow E=\left( {{2}^{2\times \frac{1}{2}}}\times {{2}^{2\times \frac{1}{2}}}\times {{2}^{\frac{1}{2}}}\times {{3}^{2\times \frac{1}{2}}}\times {{5}^{\frac{1}{2}}} \right) \\
 & \Rightarrow E=\left( 2\times 2\times {{2}^{\frac{1}{2}}}\times 3\times {{5}^{\frac{1}{2}}} \right) \\
 & \Rightarrow E=\left( 12\times {{2}^{\frac{1}{2}}}\times {{5}^{\frac{1}{2}}} \right) \\
\end{align}\]
Since 2 and 5 are not grouped and have the same exponent so we can again write them inside the square root sign and multiply them, the expression becomes:
\[\begin{align}
  & \Rightarrow E=\left( 12\times {{\left( 2\times 5 \right)}^{\frac{1}{2}}} \right) \\
 & \Rightarrow E=\left( 12\times {{10}^{\frac{1}{2}}} \right) \\
 & \Rightarrow E=\left( 12\times \sqrt{10} \right) \\
 & \therefore E=12\sqrt{10} \\
\end{align}\]
Hence, the above expression represents the simplified form of $\sqrt{1440}$.

Note: You may take into account that we haven’t asked to find the decimal representation of $\sqrt{1440}$and that is why we left the answer at \[12\sqrt{10}\]. If we are asked to further simplify it and find the decimal representation then we will use the long division method to find the square root of 10 and finally multiply it with 12 to get the answer. Remember the basic formulas of exponents and the prime factorization method.