Every surd is an irrational, but every irrational need not be a surd. Justify your Answer
Answers
Surds are subsets of irrational numbers.
If nth root of 'a' is an irratioal number and 'a' is a rational number. Then such irratioal number is called a surd.
Square root 2 is a surd
Square root 7 is a surd
Square root 5/7 is a surd
Cube root 9 is a surd
Fourth root 5 is a surd
Why???
Because under the square root or cube root or radical symbols all numbers or radicands are rationals like 2, 7, 5/7, 9, 5 etc etc. So along with radical symbols all these numbers make irrational numbers which are surds .
But ..
Square root of (3+Sq root5) is an irrational number but not a surd
Square root of ( 7- sq root3) is an irrational number but not a surd
Cube root of ( 11+ sq root 7) is an irratioal number but not a surd
Why???
Because under the radical symbols all radicands are irrational like (3+sq toot5),(7-sq root 3), (11+ sq root7) etc etc are all irrationals . So along with main radical symbols all these numbers make irrationals & surds as well.
But now
What about square root of square root of 3 ?
Since sq root of sq root of 3 = fourth root of 3 . So it will be irrational which is a surd. Because under the radical sign ( 4th root symbol) there is 3 , which is a rational.
Similarly cube root of square root of 7 = sixth root of 7 , which is an irrational which is a surd.
Conclusion : Every surd is an irrational number but every irrational number is not a surd.
Answer:
surd is an irrational root of a rational number.So now we know that surds are always irrational and they are always roots. ... On the other hand, is not a surd even though is irrational, because is not rational.
Hope its help you....