given that 2^2x+2 x 5^x-1 = 8^x x 5^2x , evaluate 10^x

We are given the equation:

\[
2^{2x} + 2 \times 5^{x-1} = 8^x \times 5^{2x}
\]

To solve this and evaluate \( 10^x \), we need to first simplify both sides of the equation.

### Step 1: Simplify the terms
Start by rewriting everything in terms of simpler bases.

- \( 2^{2x} \) is already in its simplest form.
- \( 8^x \) can be rewritten as \( (2^3)^x = 2^{3x} \).
- So, the equation becomes:

\[
2^{2x} + 2 \times 5^{x-1} = 2^{3x} \times 5^{2x}
\]

### Step 2: Simplify the equation
Now, break down the equation step by step:

\[
2^{2x} + 2 \times 5^{x-1} = 2^{3x} \times 5^{2x}
\]

To make things easier, let's deal with the second term on the left-hand side:

\[
2 \times 5^{x-1} = 2 \times \frac{5^x}{5} = \frac{2 \times 5^x}{5}
\]

Thus, the equation becomes:

\[
2^{2x} + \frac{2 \times 5^x}{5} = 2^{3x} \times 5^{2x}
\]

### Step 3: Attempt further simplification
At this point, it's quite complex to isolate terms directly, so another way to handle this is to consider values of \( x \) that might satisfy this equation by trial and error.

Let me try this next approach.It looks like there was an issue with solving the equation directly. However, we can simplify further by recognizing that powers of 2 and 5 dominate the equation.

Let's check some likely values of \( x \) manually:

### Try \( x = 1 \):
Substituting \( x = 1 \) into the original equation:

\[
2^{2(1)} + 2 \times 5^{1-1} = 8^{1} \times 5^{2(1)}
\]
This simplifies to:

\[
2^2 + 2 \times 1 = 8 \times 25
\]
\[
4 + 2 = 200
\]
which is false.

### Try \( x = 0 \):
Substituting \( x = 0 \) into the original equation:

\[
2^{2(0)} + 2 \times 5^{0-1} = 8^{0} \times 5^{2(0)}
\]
\[
1 + 2 \times \frac{1}{5} = 1 \times 1
\]
\[
1 + \frac{2}{5} = 1
\]
which is also false.

### Try \( x = 2 \):
Substituting \( x = 2 \) into the original equation:

\[
2^{2(2)} + 2 \times 5^{2-1} = 8^{2} \times 5^{2(2)}
\]
\[
2^4 + 2 \times 5 = 64 \times 625
\]
\[
16 + 10 = 40000
\]
which is also false.

Given the complexity of the expression, I suggest an alternative approach using logarithms or numerical methods to find a better solution.

Would you like more details on one of these methods, or would you like to try a specific value?

given that 2^2x + 2 x 5^x-1 = 8^x x 5^2x , evaluate 10^x

Solve the exponential equation 2^(2x) + 2 * 5^(x-1) = 8^x * 5^(2x) and evaluate 10^x. This solution involves using laws of exponents and trial and error methods. Ideal for high school students in Grades 9-12.

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