Use your knowledge of exponents to solve.

a) 1/2^x=1/(x+2)

b) 1/2^x>1/x^2

So I know that these functions are rational functions.. and I am trying to solve for x. I tried to solve them by I keep getting stuck with the exponent 2 which is the exponential function.. Help please

a) the first one is solved by inspection, x=0 is the only solution.

b) the second is solved by inspection, if x=2 the sides are equal, so then try on each side. If x=1 or x=3, is it true?

at x=1
1/2?>1 False, so x cannot be <2
at x=3
1/8?>1/9 True, so x>2

To solve the given equations involving exponents, we need to manipulate the expressions and isolate the variable x. Let's take it step by step:

a) 1/2^x = 1/(x+2)

To simplify the equation, we can express both sides with the same base:

2^-x = 1/(x+2)

Now, we can take the reciprocal on both sides to eliminate the fraction:

2^x = (x+2)

Since we have an exponential expression on one side and a polynomial expression on the other, it might be difficult to find an algebraic solution. We can use numerical or graphical methods to estimate an approximate solution or find an intersection point.

b) 1/2^x > 1/x^2

Let's simplify the equation first:

2^x < x^2

Here, we have an inequality involving exponential and polynomial functions. Similarly, finding an algebraic solution might not be possible, so we can resort to methods like numerical or graphical approaches.

To solve the equations involving exponents, we can first convert the equations into a common base. In this case, we will convert everything into base 2, as both equations involve 2 raised to some power.

a) 1/2^x = 1/(x+2)

To remove the fractions, we can multiply both sides of the equation by 2^x:

1 = 2^x / (x+2)

Next, we can multiply both sides by (x+2) to eliminate the denominator:

(x+2) = 2^x

Now, we have an equation where both sides are in the form of 2 raised to some power. We can solve this equation by trial and error or using graphical methods. One possible solution is x = 2.

b) 1/2^x > 1/x^2

First, let's multiply both sides of the equation by 2^x to eliminate the fractions:

1 > (2^x) / (x^2)

Next, let's multiply both sides by x^2 to move the terms involving x^2 to the left side:

x^2 > (2^x)

Now, we have an inequality where both sides are in the form of 2 raised to some power. We can solve this by trial and error or by observing the behavior of the two functions. One possible solution is x > 4.

Remember that these solutions are just examples, and you may need to explore further or use numerical methods to find all possible solutions.