I have two questions that needs to be solved with algebra, please help, i figured out the answers to be -4 for the first and -1 for the second but i dont know the steps solving it with algebra.
2^{x} = 1/16 Find X
(0.5)^{x} = 2 Find X
Your problem seems to be not understanding what "indices" (I would say "exponents" which is a little more specific than "indices") mean.$\displaystyle x^2$ ('x squared') means 'x times x', $\displaystyle x^3$ means 'x times x times x', etc. Further, a negative exponent means "reciprocal" or "1 over" the value. In particular $\displaystyle 2^4= 2(2)(2)(2)= 16$ so that $\displaystyle 2^{-4}= \frac{1}{16}$.
Of course, $\displaystyle 0.5= \frac{1}{2}$ so that $\displaystyle 0.5= \left(\frac{1}{2^1}\right)= 2^{-1}$ and $\displaystyle (0.5)^x= (2^{-1})^x= 2^{-x}$
So the first equation says $\displaystyle 2^x= 2^{-4}$ and comparing the two sides x= -4.
The second equation says $\displaystyle 2^{-x}= 2^1$ and comparing the two sides gives -x= 1.
(There is a technical point here- that the exponential function is "one-to-one" which means that if f(x)= f(y) then x= y. Not all functions have that property but the exponential does. All you need to know is that you can do that for exponential functions.)
Definition of exponentiation: b^m = b * b * b * … * b (b times itself m times)
For example: b^4 × b^3 =
( b × b × b × b) × ( b × b × b) = (definition of exponentiation)
b × b × b × b × b × b × b = (associative property of multiplication)
b^7 (definition of exponentiation)
If b^m=b^n
b^m/b^n=1
b^(m-n)=1
m-n=0
Hence, m=n