If `x^((3)/(2))=27, x^((5)/(2))=`

Options

A. 27
B. 81
C. 243
D. 729

Explanation

To solve the problem where \( x^{\frac{3}{2}} = 27 \) and we need to find \( x^{\frac{5}{2}} \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ x^{\frac{3}{2}} = 27 \]

### Step 2: Take the cube root of both sides
To isolate \( x \), we can take the cube root of both sides. This means we will raise both sides to the power of \( \frac{1}{3} \):
\[ \left(x^{\frac{3}{2}}\right)^{\frac{1}{3}} = 27^{\frac{1}{3}} \]

### Step 3: Simplify the left side
Using the property of exponents \( (a^m)^n = a^{m \cdot n} \), we simplify the left side:
\[ x^{\frac{3}{2} \cdot \frac{1}{3}} = 27^{\frac{1}{3}} \]
This simplifies to:
\[ x^{\frac{1}{2}} = 3 \]

### Step 4: Square both sides
Next, we square both sides to solve for \( x \):
\[ (x^{\frac{1}{2}})^2 = 3^2 \]
This gives us:
\[ x = 9 \]

### Step 5: Find \( x^{\frac{5}{2}} \)
Now that we have \( x = 9 \), we can find \( x^{\frac{5}{2}} \):
\[ x^{\frac{5}{2}} = 9^{\frac{5}{2}} \]

### Step 6: Simplify \( 9^{\frac{5}{2}} \)
We can rewrite \( 9 \) as \( 3^2 \):
\[ 9^{\frac{5}{2}} = (3^2)^{\frac{5}{2}} = 3^{2 \cdot \frac{5}{2}} = 3^5 \]

### Step 7: Calculate \( 3^5 \)
Now we calculate \( 3^5 \):
\[ 3^5 = 243 \]

### Conclusion
Thus, the value of \( x^{\frac{5}{2}} \) is:
\[ \boxed{243} \]