4by7 raise to power 12 divided by 4 by7 raise to power p is 64 by 343. Find the value of p

We are given the equation:

\[
\frac{\left( \frac{4}{7} \right)^{12}}{\left( \frac{4}{7} \right)^p} = \frac{64}{343}
\]

### Step-by-Step Solution:

1. **Simplify the left-hand side:**

   Use the law of exponents \(\frac{a^m}{a^n} = a^{m-n}\) to combine the powers of \(\frac{4}{7}\):

   \[
   \frac{\left( \frac{4}{7} \right)^{12}}{\left( \frac{4}{7} \right)^p} = \left( \frac{4}{7} \right)^{12 - p}
   \]

   So, the equation becomes:

   \[
   \left( \frac{4}{7} \right)^{12 - p} = \frac{64}{343}
   \]

2. **Express \(\frac{64}{343}\) as a power of \(\frac{4}{7}\):**

   Notice that:
   \[
   64 = 4^3 \quad \text{and} \quad 343 = 7^3
   \]

   Thus:
   \[
   \frac{64}{343} = \left( \frac{4}{7} \right)^3
   \]

3. **Equate the exponents:**

   Now, equate the exponents on both sides of the equation:

   \[
   12 - p = 3
   \]

4. **Solve for \(p\):**

   \[
   12 - p = 3
   \]
   \[
   p = 12 - 3
   \]
   \[
   p = 9
   \]

Thus, the value of \(p\) is \(9\).

---

Would you like further details or explanations on any part? Here are some related questions to explore:

1. How do exponent laws work for fractions?
2. How can we simplify expressions with powers of fractions?
3. What other methods can be used to solve for unknown exponents?
4. How do you convert numbers into powers of other numbers?
5. What are some real-world applications of exponentiation with fractions?

**Tip:** Always check if the numbers on both sides of an equation can be expressed as powers of the same base—it simplifies the solution process!

4 by 7 raised to the power of 12 divided by 4 by 7 raised to the power p is 64 by 343. Find the value of p.

Learn how to find the value of p in the equation (4/7)^12 divided by (4/7)^p equals 64/343. This solution covers key exponent rules, simplification techniques, and step-by-step calculations. Ideal for students in Grades 8-10.

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