Question

What is the order of the surd 3/4/8?
a) 3
b) 4
c) 7
d)12​

Answer 1

SOLUTION

CORRECT QUESTION

TO CHOOSE THE CORRECT OPTION

$\displaystyle \sf{ The \: order \: of \: the \: surd \: \: \sqrt[3]{ \sqrt[4]{8} } }$

a) 3

b) 4

c) 7

d) 12

CONCEPT TO BE IMPLEMENTED

If an irrational numbers can be written in the form $\sf{ \sqrt[n]{x} }$

Where x is a rational number and n is a natural number

$\sf{Then \: \: n \: \: is \: the \: order \: of \: the \: surd \: \: \sqrt[n]{x} }$

EVALUATION

Here the given surd is

$\displaystyle \sf{ \: \sqrt[3]{ \sqrt[4]{8} } }$

Now we simplify it as below

$\displaystyle \sf{ \sqrt[3]{ \sqrt[4]{8} } }$

$\displaystyle \sf{ = { \bigg( \sqrt[4]{8} \bigg)}^{ \frac{1}{3} } }$

$\displaystyle \sf{ = { \bigg( {8}^{ \frac{1}{4} } \bigg)}^{ \frac{1}{3} } }$

$\displaystyle \sf{ = { \bigg(8 \bigg)}^{ \frac{1}{4} \times \frac{1}{3} } }$

$\displaystyle \sf{ = { \bigg( {2}^{3} \bigg)}^{ \frac{1}{12} } }$

$\displaystyle \sf{ = { \bigg( {2}^{} \bigg)}^{ 3 \times \frac{1}{12} } }$

$\displaystyle \sf{ = { \bigg( {2}^{} \bigg)}^{ \frac{1}{4} } }$

$\displaystyle \sf{ = \sqrt[4]{2} }$

Comparing with $\sf{ \sqrt[n]{x} }$ we get

n = 4 , x = 2

Order of the surd = 4

FINAL ANSWER

Hence the correct option is b) 4

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