Find the value of `(1296)^(0.75)(36)^(-1)`

Answer

To solve the expression \((1296)^{0.75} \cdot (36)^{-1}\), we will break it down step by step.

### Step 1: Rewrite \(0.75\) as a fraction
We can express \(0.75\) as a fraction:
\[ 0.75 = \frac{75}{100} = \frac{3}{4} \]

**Hint:** Remember that \(0.75\) can be converted to a fraction by considering it as \(75\) out of \(100\) and simplifying.

### Step 2: Express \(1296\) as a power of \(36\)
Next, we recognize that \(1296\) is the square of \(36\):
\[ 1296 = 36^2 \]

**Hint:** If you're unsure about the relationship between numbers, you can always check by calculating \(36 \times 36\).

### Step 3: Substitute into the expression
Now we can substitute \(1296\) in the original expression:
\[ (1296)^{0.75} = (36^2)^{0.75} \]

**Hint:** When dealing with exponents, remember that \((a^m)^n = a^{m \cdot n}\).

### Step 4: Apply the exponent rule
Using the exponent rule, we can simplify:
\[ (36^2)^{0.75} = 36^{2 \cdot 0.75} = 36^{1.5} \]
Since \(1.5\) can be expressed as \(\frac{3}{2}\), we have:
\[ 36^{1.5} = 36^{\frac{3}{2}} \]

**Hint:** Multiplying the exponents is a key step when simplifying powers.

### Step 5: Rewrite \(36^{\frac{3}{2}}\)
Now, we can express \(36^{\frac{3}{2}}\) as:
\[ 36^{\frac{3}{2}} = (36^{\frac{1}{2}})^3 = (\sqrt{36})^3 \]

**Hint:** The square root of a number can be calculated easily and is often a simpler form to work with.

### Step 6: Calculate \(\sqrt{36}\)
Calculating \(\sqrt{36}\):
\[ \sqrt{36} = 6 \]

**Hint:** Knowing perfect squares can help quickly find square roots.

### Step 7: Raise to the power of \(3\)
Now we raise \(6\) to the power of \(3\):
\[ (6)^3 = 6 \cdot 6 \cdot 6 = 216 \]

### Step 8: Combine with \((36)^{-1}\)
Now we need to consider the second part of our original expression, which is \((36)^{-1}\):
\[ (36)^{-1} = \frac{1}{36} \]

### Step 9: Combine the results
Now we combine both parts:
\[ (1296)^{0.75} \cdot (36)^{-1} = 216 \cdot \frac{1}{36} \]

### Step 10: Calculate the final result
Calculating:
\[ 216 \div 36 = 6 \]

Thus, the final answer is:
\[ \boxed{6} \]