Question

How do you simplify the square root $\pm \sqrt{\dfrac{16}{49}}$?

Answer 1

Hint: We will look at the definition of a square root. We will look at some examples, that is, square roots of some numbers. Then we will discuss the prime factorization method of finding square roots. We will use this method to find the square root of the numerator and the denominator. Then we will obtain the simplification of the given term.

Complete step by step answer:
We define the square root of a number to be a value such that, when multiplied to itself, it gives back the given number as the product. So, for example, we know that $2\times 2=4$. Here, 2 is multiplied to itself to obtain the number 4. So, 2 is the square root of 4 and 4 is the square of 2. Similarly, we know $3\times 3=9$. So, 3 is the square root of 9 and 9 is the square of 3.
We can find the square root of a number by the prime factorization method. In this method, we write the prime factorization of the given number and observe that a set of primes is being multiplied two times. Both these sets contain the same prime factors, hence their product is the same and this product is the square root of the given number. For example, consider the number 36. The prime factorization of 36 is as follows,
$\begin{align}
  & 2\left| \!{\underline {\,
  36 \,}} \right. \\
 & 2\left| \!{\underline {\,
  18 \,}} \right. \\
 & 3\left| \!{\underline {\,
  9 \,}} \right. \\
 & 3\left| \!{\underline {\,
  3 \,}} \right. \\
 & \left| \!{\underline {\,
  1 \,}} \right. \\
\end{align}$
We can see that 2 and 3 are the prime factors and they are being repeated twice. Therefore, the square root of 36 is $2\times 3=6$. That means, $\sqrt{36}=6$ or $6\times 6=36$.
Now, the given term is $\pm \sqrt{\dfrac{16}{49}}$. We can write the numerator as
$\begin{align}
  & 16=2\times 2\times 2\times 2 \\
 & \therefore 16=4\times 4 \\
\end{align}$
So, 4 is the square root of 16. Similarly, the denominator is, $49=7\times 7$. So, the square root of 49 is 7.

Hence, the given term can be simplified as
$\pm \sqrt{\dfrac{16}{49}}=\pm \dfrac{4}{7}$.

Note: For larger numbers, we can find the square root using the long division method. We should be familiar with the concept of squares and square roots since they come up in a lot of calculations. These concepts are also valid for expressions. If we multiply an algebraic expression to itself, the product obtained in the square of that expression.