Simplifying radicals is the process of manipulating a radical expression into a simpler or alternate form. Generally speaking, it is the process of simplifying expressions applied to radicals.
A radical is a number that has a fraction as its exponent:
Since they are exponents, radicals can be simplified using rules of exponents.
The square root of a positive integer that is not a perfect square is always an irrational number. The decimal representation of such a number loses precision when it is rounded, and it is time-consuming to compute without the aid of a calculator. Instead of using decimal representation, the standard way to write such a number is to use simplified radical form, which involves writing the radical with no perfect squares as factors of the number under the root symbol.
The process for putting a square root into simplified radical form involves finding perfect square factors and then applying the identity \(\sqrt=\sqrt\times\sqrt\), which allows us to take the root of the perfect square factors.
See SolutionSimplify \(\sqrt \).
Since \( 12 = 2 \times 2 \times 3= 2^2 \times 3 \), we can rewrite it as \( \sqrt = \sqrt < 2^2 \times 3 >= \sqrt \times \sqrt = 2 \sqrt.\) \(_\square\)
\(4\) is a perfect square factor of \(72\), and \(9\) is a perfect square factor of \(72\).
For the sake of this process, it is more efficient to find the largest perfect square factor of \(72\). As shown below, \(4\times 9=36\) is the largest perfect square factor of \(72:\)
\[\begin \sqrt&=\sqrt \\ &=\sqrt\times\sqrt \\ &=6\sqrt. \end\]
Therefore, simplified form of \(\sqrt\) is \(6\sqrt.\ _\square\)
Note: When a number is placed to the left of a square root symbol, multiplication is implied. \(“\, 6\sqrt\, ”\) is read as \(“\, 6\) times the square root of \(2.\, ”\)
Similarly, roots of higher degree (cube roots, fourth roots, etc.) are simplified when they have no factors under the radical that are perfect powers of the same degree as the radical.
Simplify \(\sqrt[3] < a^2 b^ 4 >\).
Notice that we have \( b^3 \), which is a cube factor in the radicand. Hence, we can pull it out to obtain
\[ \sqrt[3] < a^2 b^4 >= b \sqrt[3] < a^2b>.\ _\square\]