What is #3 sqrt(x^2/y)# in exponential form?

3 Answers

#(3x)/y^(1/2)#

Explanation:

#sqrt(x^2)# can be written as #x#, but y does not have a square root.

#3xy^(-1/2)#, but it is better not to have negative exponents.

#(3x)/y^(1/2)#

If you wanted to rationalize the denominator:

#(3x)/y^(1/2) xx y^(1/2)/y^(1/2)#

#(3xy^(1/2))/y#

Jul 1, 2016

Exponent form: #+-3xy^(-1/2)#

Explanation:

Here, #y>0# for #3sqrt(x^2/y)# to be real.
The form with exponents is #3((x^2)^(1/2))/y^(1/2)=3x/y^(1/2)#

#=3xy^(-1/2)#

Jul 3, 2016

#3sqrt(x^2/y) = 3abs(x)y^(-1/2)# for #y in (0,oo)#

Explanation:

Considering only Real valued square roots, we require:

#x^2/y >= 0#

Since #x^2>=0# for any Real #x#, this amounts to #y > 0#.

Note that #sqrt(x^2) = abs(x)# for any Real #x#. The square root sign denotes the principal square root, which in the case of Real square roots is the non-negative one.

Note that if #a >= 0# and #b > 0# then #sqrt(a/b) = sqrt(a)/sqrt(b)#

So we find:

#3sqrt(x^2/y) = 3(sqrt(x^2))/(sqrt(y)) = 3abs(x)/y^(1/2) = 3abs(x)y^(-1/2)#