Question

underroot x equal to x to the power 1/2 kaise hota h EXPLAIN because I'm solving maths application of integrals ​

Answer 1

Step-by-step explanation:

Let's assume

${x}^{2} = y = {y}^{1} \: \: \: (1) \\ x \times x = y \\ x ^{(1 + 1)} = {y}^{1} \\ {x}^{2} = y ^{1} \\ dividing \: both \: exponents \: by \: 2 \\ {x}^{ \frac{2}{2} } = {y}^{ \frac{1}{2} } \\ {x}^{1} = y ^{ \frac{1}{2} } \\ x = {y}^{ \frac{1}{2} } \\ but \\ x = \sqrt{y} \: \: \: (from \: eqn \: 1) \\ hence \: \sqrt{y } = {y}^{ \frac{1}{2} }$

Answer 2

the notation √ is a concept used to denote some particular type of numbers.

we know that √4 means we need to find a number which when multiplied by itself two times gives 4. the number is 2 (2 x 2 = 4)

we call 2 as square root of 4 i.e.

√4 = 2

similarly 3√8 means a number which when multiplied by itself three times, gives 8. the number is 2 (2 x 2 x 2 = 8)

we call 2 as cuberoot of 8 i.e.

3√8 = 2

as we discussed above

√4 = 2

in exponent form let us 4 raised to the power x is 2

4^x = 2

(2^2)^x = 2

2^(2x) = 2^1

on comparing the exponents

2x = 1

x = 1/2

thus the notation √4 is equivalent to 4^(1/2) in exponent form i.e.

√4 = 4^(1/2)

in general

√x = x^(1/2)

It is important to know that if there is a small number just before the √ mark like

$\sqrt[3]{27}$

then exponent form is not (1/2). It will be (1/3)

i.e. (27)^(1/3)

as a general rule if nothing preceded √ sign, exponent is (1/2), if some number n preceeds √ sign, the exponent is (1/n).

we also need to understand the difference between

3*√27 and

$\sqrt[3]{27}$

the former is 3 x (27)^(1/2), the latter is (27)^(1/3)

hope I was clear enough.