Question

State, with the reason, which of the following are surds and which are not:
1) $\sqrt{180}$
2) $\sqrt[4]{27}$
3) $\sqrt[5]{128}$
4) $\sqrt[3]{64}$
5) $\sqrt[3]{25}\cdot \sqrt[3]{40}$
6) $\sqrt[3]{-125}$
7) $\sqrt{\pi }$
8) $\sqrt{3+\sqrt{2}}$

Answer 1

Hint: In this problem to decide which is surd or not. We first define what is surds and some properties of surds then by definition we will decide which is surd or not. Or, we will factorise some of the terms and if on factorization, we get any natural number out of the root, then the number will not be surd and if the number remains under root then it will be surd.

Complete step-by-step answer:
Surds- let x be the number and n be a positive integer such that $\sqrt[n]{x}$ is not a perfect root of x. Then the number $\sqrt[n]{x}$ is called the surd of order n.
Properties of surds:
a) $\sqrt[n]{a}\cdot \sqrt[n]{b}=\sqrt[n]{ab}$
b) $\sqrt[n]{\dfrac{a}{b}}=\dfrac{\sqrt[n]{a}}{\sqrt[n]{b}}$

1) $\sqrt{180}$ is surds. Since 180 does not have a perfect square root.
2) $\sqrt[4]{27}=\sqrt[4]{{{3}^{3}}}$
$\sqrt[4]{27}=3\sqrt[{}]{3}$ is surds. Since 27 does not have a perfect fourth root.
3) $\sqrt[5]{128}$ is surds. Since 128 does not have a perfect 5th root.
4) $\sqrt[3]{64}$is not surds. Since 64 has a perfect cube root which 4.
5) $\sqrt[3]{25}\cdot \sqrt[3]{40}$=$\sqrt[3]{25\cdot 40}=\sqrt[3]{1000}$
$\sqrt[3]{{{10}^{3}}}=10$ by using property of surds.
    $\sqrt[3]{25}\cdot \sqrt[3]{40}$ is not a surds. Since 1000 has a perfect cube root which is 10.
6) $\sqrt[3]{-125}=-\sqrt[3]{125}$.
 $\sqrt[3]{-125}$ is not a surds. Since 125 have a perfect cube root which is 5.
7) $\sqrt{\pi }$ is surds. Since $\pi $ does not have a perfect square root.
8) $\sqrt{3+\sqrt{2}}$ is surds. Since $\sqrt{2}$ is surds and $3+\sqrt{2}$ does not have perfect square roots.

Note: Always remember that, Surds are the numbers that include a square root, cube root or any other root symbol. Surds are used to write irrational numbers precisely. For example, ${{\left( 6 \right)}^{\dfrac{1}{7}}}$ , $\sqrt{78}$
In this problem, students should remember the following points regarding surds.
1) Every rational is not surds.
2) Every irrational is surds.
3) Product of two surds cannot be surds
e.g $\sqrt{2}\cdot \sqrt{2}=2$ which is not a surds.
4) Sum of two surds are surds