2 by 7 raise to power 8x divided by 4 by 49 multiplied by 2 by 7 raise to power3 is equal to 2 by 7 raise to power 4x-7. Find value of x

We are given the equation:

\[
\frac{\left(\frac{2}{7}\right)^{8x}}{\frac{4}{49}} \times \left(\frac{2}{7}\right)^3 = \left(\frac{2}{7}\right)^{4x-7}
\]

### Step 1: Simplify the fractions
Notice that \( \frac{4}{49} = \left(\frac{2}{7}\right)^2 \), so we can rewrite the equation as:

\[
\frac{\left(\frac{2}{7}\right)^{8x}}{\left(\frac{2}{7}\right)^2} \times \left(\frac{2}{7}\right)^3 = \left(\frac{2}{7}\right)^{4x-7}
\]

### Step 2: Use the properties of exponents
Using the property of exponents \( \frac{a^m}{a^n} = a^{m-n} \), we can simplify the left side:

\[
\left(\frac{2}{7}\right)^{8x-2} \times \left(\frac{2}{7}\right)^3 = \left(\frac{2}{7}\right)^{4x-7}
\]

Now apply the exponent addition property \( a^m \times a^n = a^{m+n} \) on the left side:

\[
\left(\frac{2}{7}\right)^{(8x - 2) + 3} = \left(\frac{2}{7}\right)^{4x-7}
\]

Simplifying the exponent on the left side:

\[
\left(\frac{2}{7}\right)^{8x + 1} = \left(\frac{2}{7}\right)^{4x-7}
\]

### Step 3: Set the exponents equal
Since the bases are the same, we can set the exponents equal to each other:

\[
8x + 1 = 4x - 7
\]

### Step 4: Solve for \(x\)
Subtract \(4x\) from both sides:

\[
4x + 1 = -7
\]

Subtract 1 from both sides:

\[
4x = -8
\]

Divide by 4:

\[
x = -2
\]

Thus, the value of \(x\) is \( \boxed{-2} \).

Would you like more details or have any questions?

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Here are 5 related questions:
1. What are the properties of exponents?
2. How do you simplify fractions with variables in the exponents?
3. Can you explain how to solve exponential equations with different bases?
4. What are common mistakes to avoid when working with fractional exponents?
5. How does changing the base of an exponential expression affect the solution?

**Tip:** When working with exponential equations, always make sure the bases are the same before comparing exponents.

Learn how to solve an exponential equation involving fractional exponents: (2/7)^(8x) / (4/49) * (2/7)^3 = (2/7)^(4x-7). This problem covers key algebraic concepts such as properties of exponents and solving exponential equations step-by-step. Suitable for students in Grades 9-12.

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