Question

5^-x - 5^ x-2=24 4^x+2^x-3=20 3^x-2-3^x=72

Answer 1

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Answer 2

You have presented a system of three exponential equations. Let's solve them step by step:

Equation 1: $5^{-x} - 5^{x-2} = 24$

Equation 2: $4^{x} + 2^{x-3} = 20$

Equation 3: $3^{x-2} - 3^{x} = 72$

Let's start with Equation 1:

Equation 1: $5^{-x} - 5^{x-2} = 24$

First, let's rewrite the equation in a more manageable form:

$\frac{1}{5^x} - \frac{5^{x}}{25} = 24$

Now, let's deal with the fractions and find a common denominator:

$\frac{1}{5^x} - \frac{1}{5^x} = 24$

This simplifies to:

$0 = 24$

Since this is not possible, there is no solution for Equation 1.

Now, let's move on to Equation 2:

Equation 2: $4^{x} + 2^{x-3} = 20$

First, let's try to get both terms to have the same base. Notice that $4 = 2^2$, so we can rewrite the equation as:

$(2^2)^{x} + 2^{x-3} = 20$

Using the properties of exponents, we get:

$2^{2x} + 2^{x-3} = 20$

Now, we can combine the terms with the same base:

$2^{2x} + 2^{x} \cdot 2^{-3} = 20$

$2^{2x} + \frac{1}{8} \cdot 2^{x} = 20$

$2^{2x} + \frac{2^{x}}{8} = 20$

$2^{2x} + 2^{x-3} = 20$

At this point, notice that the equation is similar to the original equation, so we can see that there's no solution for Equation 2.

Finally, let's move on to Equation 3:

Equation 3: $3^{x-2} - 3^{x} = 72$

We can rewrite the equation as:

$3^{x-2} - 3^{x} = 3^{x} \cdot (1 - 3^2) = 72$

Simplifying the right side:

$3^{x} \cdot (-8) = 72$

$-8 \cdot 3^{x} = 72$

Dividing both sides by -8:

$3^{x} = -9$

Since the left side represents an exponential function and the right side is negative, there is no real solution for Equation 3.

In summary, after analyzing the three equations, it appears that there are no real solutions to the given system of exponential equations.