Answer:
Explanation:
To solve the equation 5^(x+1) + 5^(2-x) = 5^3 + 1, we need to find the value of x that satisfies this equation.
First, let's simplify the equation using the properties of exponents.
We know that 5^3 is equal to 5 * 5 * 5, which is 125. So the equation becomes:
5^(x+1) + 5^(2-x) = 125 + 1
Next, let's work on simplifying the left side of the equation.
The rule for adding exponents with the same base states that when we add exponents with the same base, we multiply the bases and keep the exponent the same.
So, we can rewrite the left side of the equation as:
5^x * 5^1 + 5^2 * 5^(-x)
Simplifying further, we have:
5^x * 5 + 5^2 / 5^x
Now, let's simplify the right side of the equation.
125 + 1 is equal to 126.
So, the equation becomes:
5^x * 5 + 5^2 / 5^x = 126
To solve for x, let's combine the terms with the same base.
We have 5^x * 5 and 5^2 / 5^x.
Since the bases are the same (5), we can add the exponents and rewrite the equation as:
5^x+1 + 5^2-x = 126
Now, let's simplify the equation further:
5^x+1 + 5^2-x = 126
25 * 5^x / 5 + 5^2 / 5^x = 126
25 * 5^x / 5 + 25 / 5^x = 126
5^x / 5 + 25 / 5^x = 126
5^x/5 + 5^2/x = 126
Now, let's multiply both sides of the equation by 5 to eliminate the denominators:
5 * (5^x / 5) + 5 * (5^2/x) = 5 * 126
5^x + 5^2 / x = 630
Now, we have an equation without any fractions.
To further simplify the equation, we can rewrite 5^2 as 25:
5^x + 25 / x = 630
Now, let's multiply both sides of the equation by x to eliminate the denominator:
x * (5^x + 25 / x) = 630 * x
x * 5^x + 25 = 630x
Finally, we can subtract 630x from both sides of the equation to isolate the variable:
x * 5^x - 630x + 25 = 0
At this point, solving the equation for x algebraically becomes quite difficult. It may be more practical to use numerical methods or a graphing calculator to find the approximate solutions.