$${\frac{{\left|{\mathtt{5}}{\mathtt{\,\small\textbf+\,}}{\sqrt[{{\mathtt{{\mathtt{X}}}}}]{-{\mathtt{177\,147}}}}\right|}}{\left({\mathtt{2}}{\mathtt{\,\small\textbf+\,}}{\sqrt{{\mathtt{5}}}}\right)}} = \left({\mathtt{2}}{\mathtt{\,\times\,}}{\sqrt{{\mathtt{5}}}}{\mathtt{\,-\,}}{\mathtt{4}}\right)$$

Guest Mar 13, 2015

#2**+5 **

Mutiply both sides by the denominator on the left

l 5 +^{ x}√-177147 l = (2√5 - 4) (√5 + 2)

l 5 +^{ x}√-177147 l = 10 - 8

l 5 +^{ x}√-177147 l = 2

This says that either

-(5 + ^{x}√-177147 ) = 2 or (5 + ^{x}√-177147 ) = 2

Working with the first we have

- ^{x}√-177147 = 7

^{x}√-177147 = -7

-177177 = (-7)^x .....and this has no real solutions

Working with the second, we have

^{x}√-177147 = -3

(-3)^x = -177147

Note that, we cannot take the log of both sides here because log(-3)^x would be undefined as well as log(-177147)

However.......if this has an integer solution, x must be an odd number....

So we can write

(-1)^x (3)^x = -177147

And, if x is an odd integer, (-1)^x = -1.....so we have

(-1) * (3)^x = -177147 divide both sides by -1

(3)^x = 177147

Now...take the log of both sides....and using a log property, we have

x log3 = log 177147 divide both sides by log 3

x = log 177147 / log 3 = 11

CPhill
Mar 14, 2015

#1**+5 **

first multiply both sides by the denominator on the right.

Then subtract 5 from both sides.

This should give you a radical to the x^{th} root of -177147 = -3

Then take the log of both sides. ..and multiply by -1.

log 177147=log 3^{x}

5.248=x log 3

x= 11

Guest Mar 13, 2015

#2**+5 **

Best Answer

Mutiply both sides by the denominator on the left

l 5 +^{ x}√-177147 l = (2√5 - 4) (√5 + 2)

l 5 +^{ x}√-177147 l = 10 - 8

l 5 +^{ x}√-177147 l = 2

This says that either

-(5 + ^{x}√-177147 ) = 2 or (5 + ^{x}√-177147 ) = 2

Working with the first we have

- ^{x}√-177147 = 7

^{x}√-177147 = -7

-177177 = (-7)^x .....and this has no real solutions

Working with the second, we have

^{x}√-177147 = -3

(-3)^x = -177147

Note that, we cannot take the log of both sides here because log(-3)^x would be undefined as well as log(-177147)

However.......if this has an integer solution, x must be an odd number....

So we can write

(-1)^x (3)^x = -177147

And, if x is an odd integer, (-1)^x = -1.....so we have

(-1) * (3)^x = -177147 divide both sides by -1

(3)^x = 177147

Now...take the log of both sides....and using a log property, we have

x log3 = log 177147 divide both sides by log 3

x = log 177147 / log 3 = 11

CPhill
Mar 14, 2015