Simplify: `((6.25)^(1/2) xx (0.0144)^(1/2)+ 1)/((0.027)^(1/3)xx (81)^(1/4))`

Options

A. 0.14
B. 1.4
C. 1
D. `1.bar4`

Answer

To simplify the expression \(\frac{(6.25)^{1/2} \times (0.0144)^{1/2} + 1}{(0.027)^{1/3} \times (81)^{1/4}}\), we will follow these steps:

### Step 1: Simplify the Numerator
1. **Calculate \((6.25)^{1/2}\)**:
- We know that \(6.25 = (2.5)^2\).
- Therefore, \((6.25)^{1/2} = (2.5)^{2 \times \frac{1}{2}} = 2.5\).

2. **Calculate \((0.0144)^{1/2}\)**:
- We know that \(0.0144 = (0.12)^2\).
- Therefore, \((0.0144)^{1/2} = (0.12)^{2 \times \frac{1}{2}} = 0.12\).

3. **Combine the results**:
- Now, we can combine these results:
\[ (6.25)^{1/2} \times (0.0144)^{1/2} = 2.5 \times 0.12 = 0.3. \]
- Adding 1 gives us:
\[ 0.3 + 1 = 1.3. \]

### Step 2: Simplify the Denominator
1. **Calculate \((0.027)^{1/3}\)**:
- We know that \(0.027 = (0.3)^3\).
- Therefore, \((0.027)^{1/3} = (0.3)^{3 \times \frac{1}{3}} = 0.3\).

2. **Calculate \((81)^{1/4}\)**:
- We know that \(81 = (3^4)\).
- Therefore, \((81)^{1/4} = (3^{4})^{\frac{1}{4}} = 3\).

3. **Combine the results**:
- Now we can combine these results:
\[ (0.027)^{1/3} \times (81)^{1/4} = 0.3 \times 3 = 0.9. \]

### Step 3: Combine Numerator and Denominator
Now we have:
\[ \frac{1.3}{0.9}. \]

### Step 4: Simplify the Fraction
1. **Divide**:
- To simplify \(\frac{1.3}{0.9}\), we can multiply both the numerator and the denominator by 10 to eliminate the decimal:
\[ \frac{1.3 \times 10}{0.9 \times 10} = \frac{13}{9}. \]

### Final Result
The simplified form of the expression is:
\[ \frac{13}{9} \approx 1.444\ldots \]
This can be approximated as \(1.44\) when rounded.