Question

How do you simplify the square root of \[30\] - square root of \[3\]?

Answer 1

Hint: We first prime factorize the given number and then, apply the properties of indices. Here, the property to be used is that when a product of some numbers is raised to some power, then it is equal to the product of all the factors of that product raised to that same power. Then, we need to take some terms common, which further contribute in simplifying the expression. Having done this, we evaluate each of the prime factor terms to arrive at the simplified form of the given expression.

Complete step-by-step solution:
The given expression is \[\sqrt{30}-\sqrt{3}\] . Let us assume,
\[x=\sqrt{30}-\sqrt{3}\]
We can observe that, \[30\] is a composite number and \[3\] is a prime number. Therefore, we need to prime factorize \[30\].
$30=2\times 3\times 5$
After performing the prime factorization, we can clearly see that the composite number \[30\]is the product of the three prime numbers \[2\] , \[3\] and \[5\] .
\[x=\surd \left( 2\times 3\times 5 \right)-\surd 3\]
We now use the property of indices, which states that,
\[{{(a\times b\times c)}^{n}}=({{a}^{n}})\times ({{b}^{n}})\times ({{c}^{n}})\]
Here, \[a=2\] , \[b=3\] , \[c=5\] and \[n=\dfrac{1}{2}\] . Therefore, writing \[x\] in the above form, we get,
\[x=\left( \left( \sqrt{2} \right)\times \left( \sqrt{3} \right)\times \left( \sqrt{5} \right) \right)-\sqrt{3}\]
We know that square root is equivalent to \[\dfrac{1}{2}th\] root. Therefore, the given expression can be rewritten as,
\[x=\left( ({{2}^{\dfrac{1}{2}}})\times ({{3}^{\dfrac{1}{2}}})\times ({{5}^{\dfrac{1}{2}}}) \right)-({{3}^{\dfrac{1}{2}}})\]
Taking \[{{3}^{\dfrac{1}{2}}}\]common from the terms, we get
\[x=({{3}^{\dfrac{1}{2}}})\times \left( ({{2}^{\dfrac{1}{2}}})\times ({{5}^{\dfrac{1}{2}}})-(1) \right)\] .... equation 1
We now need to evaluate the \[\dfrac{1}{2}th\] power of each of the numbers \[2\] , \[3\] and \[5\] .We start off with the \[\dfrac{1}{2}th\] power of \[3\] .
\[{{3}^{\dfrac{1}{2}}}=1.732\]
Then, we evaluate the \[\dfrac{1}{2}th\] power of 2.
\[{{2}^{\dfrac{1}{2}}}=1.414\]
Finally, we evaluate the \[\dfrac{1}{2}th\] power of 5.
\[{{5}^{\dfrac{1}{2}}}=2.2361\]
Putting the values of these \[\dfrac{1}{2}th\] powers of the numbers in equation 1, we get,
\[x=1.732\times \left( 1.414\times 2.2361-1 \right)\]
After multiplying \[1.414\times 2.2361\], we get,
\[x=1.732\times \left( 3.1618-1 \right)\]
Subtracting \[3.1618-1\], we get
\[x=1.732\times 2.1618\]
Finally, multiplying \[1.732\times 2.1618\] , we get,
\[x=3.7442\]
Therefore, we can conclude that the given expression in the question can be simplified to \[3.7442\].

Note: We must be careful while breaking down a composite number into its factors to take common terms. We must not break it down into all its prime factors unless required, as this would create confusions. The problem could also have been solved by directly writing the values of $\sqrt{30}$ and $\sqrt{3}$ but, there would have been a little error in the final answer as we have started rounding off in the first place.