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Question

Question: How do you simplify \(\sqrt{-1256}\)?...

How do you simplify 1256\sqrt{-1256}?

Explanation

Solution

In the above question, we have been given a square root expression to be simplified. To simplify it, we need to use the law of radicals which is given by ab=ab\sqrt{ab}=\sqrt{a}\sqrt{b} so that the given expression reduces to 11256\sqrt{-1}\sqrt{1256}. Then, since i=1i=\sqrt{-1}, the given expression will become 1256i\sqrt{1256}i. Then we need to simplify the expression 1256\sqrt{1256} by writing the prime factorisation of the number 12561256. From the prime factorisation, we can take out the highest perfect square from the square root to get the final simplified expression.

Complete step-by-step answer:
The expression given in the above question is
1256\Rightarrow \sqrt{-1256}
Now, using the law of radicals given by ab=ab\sqrt{ab}=\sqrt{a}\sqrt{b}, we can write the above expression as
11256\Rightarrow \sqrt{-1}\sqrt{1256}
Now, we know that 1\sqrt{-1} is a complex number which is equal to ii so that we can put 1=1\sqrt{-1}=1 in the above expression to get
1256i.......(i)\Rightarrow \sqrt{1256}i.......\left( i \right)
Now, in order to further simplify the above 0065pression, we have to write the prime factorization for the number 12561256, as shown below.

& 2\left| \\!{\underline {\, 1256 \,}} \right. \\\ & 2\left| \\!{\underline {\, 628 \,}} \right. \\\ & 2\left| \\!{\underline {\, 314 \,}} \right. \\\ & \text{ }\left| \\!{\underline {\, 157 \,}} \right. \\\ \end{aligned}$$ From the above, we got the prime factorization of the number $1256$ as $$\begin{aligned} & \Rightarrow 1256=2\times 2\times 2\times 157 \\\ & \Rightarrow 1256=4\times 314 \\\ \end{aligned}$$ Substituting this in (i) we get $\Rightarrow \sqrt{4\times 314}i$ Again using the law of radicals given by $\sqrt{ab}=\sqrt{a}\sqrt{b}$, we can write the above expression as $$\Rightarrow \sqrt{4}\times \sqrt{314}i$$ Now, we know that $$\sqrt{4}=2$$. Putting this above we get $\Rightarrow 2\sqrt{314}i$ Hence, the given expression has been finally simplified to $2\sqrt{314}i$. **Note:** We might argue as to why we haven’t substituted $\sqrt{4}=\pm 2$ to get two simplified expressions, one positive and the other negative. The answer to this question is that the square root of a number is always positive. If we have the equation ${{x}^{2}}=4$, then the solution to this equation will be both $2$ and $-2$. But the value of $\sqrt{4}$ is $2$ only.