https://r2.erweima.ai/mathbot/image/1726538372730pags0wj7.jpg

The expression provided in the image is quite complex, and I'll break it down step by step. The goal is to evaluate the numerical value of \( J(x) \) when \( x = 3 \) and show that the result is 27.

### Given Expression:
\[
J(x) = \left( \frac{x^{a+1}}{x^a} \right)^2 \div \left( \frac{((x^a)^{10})^{2/2} \cdot x^3}{(x^5 \cdot x^a)^{2/a}} \right)
\]

### Step 1: Simplify each part
#### Left Side: \( \left( \frac{x^{a+1}}{x^a} \right)^2 \)
This is a straightforward simplification:
\[
\frac{x^{a+1}}{x^a} = x^{a+1-a} = x^1 = x
\]
Thus, this becomes:
\[
(x)^2 = x^2
\]

#### Right Side: Simplify the denominator
The right-hand side involves a complex fraction. Let's first simplify \( (x^a)^{10} \) and other terms.

1. \( ((x^a)^{10})^{2/2} = (x^{a \cdot 10}) = x^{10a} \)
   
2. \( x^3 \) remains as it is.

Thus, the numerator of the second fraction becomes:
\[
x^{10a} \cdot x^3 = x^{10a + 3}
\]

Now for the denominator:
\[
(x^5 \cdot x^a)^{2/a} = (x^{5+a})^{2/a} = x^{\frac{2(5+a)}{a}} = x^{\frac{10+2a}{a}}
\]

Thus, the denominator becomes:
\[
x^{\frac{10 + 2a}{a}}
\]

### Step 2: Combine the simplified expressions
Now we have the simplified version of \( J(x) \):

\[
J(x) = \frac{x^2}{\frac{x^{10a+3}}{x^{\frac{10 + 2a}{a}}}}
\]

### Step 3: Substituting \( x = 3 \)
We are told to evaluate the expression for \( x = 3 \). Let's compute this expression step by step by first simplifying the exponent expressions and then plugging in the value for \( x \).

I'll compute this now.The simplified expression for \( J(x) \) when substituting \( x = 3 \) is:

\[
J(3) = 3^{\frac{-10a^2 + a + 10}{a}}
\]

### Step 4: Find the specific value of \( a \)
For the expression to evaluate to 27 (since \( 27 = 3^3 \)), we need to solve for \( a \) in the equation:

\[
3^{\frac{-10a^2 + a + 10}{a}} = 27
\]

Since \( 27 = 3^3 \), we can equate the exponents:
\[
\frac{-10a^2 + a + 10}{a} = 3
\]

I'll solve this equation for \( a \).It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like me to explain the process further or attempt another approach? Let me know if you have any specific questions about the steps.

Here are five related questions to expand on the topic:
1. How do you simplify exponents when dividing powers with the same base?
2. How can logarithms help solve exponential equations like the one here?
3. What are common techniques to simplify complex algebraic fractions?
4. How do you approach solving for variables in non-linear equations?
5. What are the properties of exponents and how do they affect equations like this?

**Tip**: Always simplify terms step-by-step before substituting numerical values to avoid mistakes when handling complex expressions.

Demuestra que el valor numérico de J(x) = ((x^(a+1))/(x^a))^2 / (((x^a)^10)^(2/2) * x^3 / (x^5 * x^a)^(2/a)) cuando x = 3, es 27.

Step-by-step simplification of the algebraic expression J(x) = ((x^(a+1))/(x^a))^2 / (((x^a)^10)^(2/2) * x^3 / (x^5 * x^a)^(2/a)) to prove that the result for x=3 is 27. This problem focuses on exponentiation and algebraic fractions, suitable for students in Grades 10-12.

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