(
9
4
​
)
3
×(
7
4
​
)
3

(
7
4
​
)
1
×(
3
2
​
)
2

​

To simplify the given expression:

\[
\left(\frac{9}{4}\right)^3 \times \left(\frac{7}{4}\right)^3 \times \left(\frac{7}{4}\right) \times \left(\frac{3}{2}\right)^2
\]

We'll simplify this step-by-step:

### Step 1: Combine the like terms
Since \(\left(\frac{7}{4}\right)\) appears in both parts, we can combine their exponents.

\[
\left(\frac{7}{4}\right)^3 \times \left(\frac{7}{4}\right) = \left(\frac{7}{4}\right)^{3+1} = \left(\frac{7}{4}\right)^4
\]

### Step 2: Simplify the entire expression
\[
\left(\frac{9}{4}\right)^3 \times \left(\frac{7}{4}\right)^4 \times \left(\frac{3}{2}\right)^2
\]

### Step 3: Calculate each term individually

1. \(\left(\frac{9}{4}\right)^3 = \frac{9^3}{4^3} = \frac{729}{64}\)
2. \(\left(\frac{7}{4}\right)^4 = \frac{7^4}{4^4} = \frac{2401}{256}\)
3. \(\left(\frac{3}{2}\right)^2 = \frac{3^2}{2^2} = \frac{9}{4}\)

### Step 4: Combine all fractions
Now multiply all the fractions together:

\[
\frac{729}{64} \times \frac{2401}{256} \times \frac{9}{4}
\]

First, let's multiply the numerators together:

\[
729 \times 2401 \times 9 = 15746409
\]

And the denominators:

\[
64 \times 256 \times 4 = 65536
\]

So, the simplified expression is:

\[
\frac{15746409}{65536}
\]

This fraction is already in its simplest form.

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. How do you multiply fractions with different denominators?
2. What is the process to simplify a fraction?
3. Can you explain the rule for multiplying exponents with the same base?
4. How do you calculate large exponents efficiently?
5. What are some common mistakes when working with exponents?
6. How do you simplify complex fractional expressions?
7. What is the significance of keeping fractions in simplest form?
8. How can you check if a fraction is in its simplest form?

**Tip:** When multiplying fractions, always simplify before performing the multiplication to make calculations easier.

Learn how to simplify the expression involving exponents and fractions: (9/4)^3 * (7/4)^3 * (7/4) * (3/2)^2. This detailed step-by-step solution covers exponentiation and fraction multiplication, suitable for high school students (Grades 9-12).

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