SURDS AND INDICES

How do I solve 4x−2x−2=0?

4x−2x−2=0

2^2X-2^X-2=0

Assume 2^x=t

then

above equation become

t²-t-2=0

t²-2t+t-2=0

from this

t=2 and t=-1

but 2^x=t

2^x=2

x=1 

and

2^x=-1

for above condition x value not possible

hence only solution x=1

How can I simplify ab√√ab√3 with exponent properties?

y=√ab/∛a∛b

y=√a√b/∛a∛b     because √ab=√a√b

y=a^(1/2-1/3)×b^(1/2-1/3)  because a^x/a^y=a^x-y

y=a^(5/6)×b^(5/6)      because a^x×b^x=(ab)^x

y=(ab)^(5/6)

If the value of (a + 1/a) is equal to 3, then what will be the value of a2 + 1/a2?

a+1/a=3

squering both sides

a²+1/a²+2×a×(1/a)=9

a²+1/a²+2=9  use formula (x+y)²=x²+y²+2xy

a²+1/a²=7

If (8x²)^8-r(2x)^-r=2^ax^{b   then find a and b in terms of r ?}
(8x²)^8-r(2x)^-r=2^ax^b

(2^³x²)^8-r(2x)^-r=2^ax^b

2^3(8-r)x^2(8-r)=2^ax^b

2^(24-3r)x^(16-2r)=2^ax^b

^{now compare the power of 2 and x}

a=(24-3r)

and b=16-2r

IF (8-√18)/√2=a+b√2 then find a and b ?
we have to eliminate √2 denominator.

for this multiply and divide by √2

i.e.

(8-√18)/√2=(8-√18)×√2/(√2×√2) 

                   =(8-√18)×√2/2

                   =(8√2-√18√2)/2

                   =(8√2-√36)/2

                   =(8√2-2√3)/2

                    =4√2-√9

                      =-3+4√2

a+b√2=-3+4√2

a=-3 and b=4
What is the value of 22014−22013

2^2014 - 2^2013

= 2^(2013+1) - 2^2013

= 2^2013 (2^1- 1)

= 2^ 2013 * (2–1)

= 2^ 2013 * 1

= 2^2013