The area of a square is `2^(4) xx 3^(6) xx 5^(10)` sq. cm .Find the length of its diagonal ( in cm ) .

Options

A. `2^(2)xx3^(3) xx5^(6) .sqrt(2)`
B. `2^(2) xx3^(4) xx5^(5) .sqrt(2)`
C. `2^(2)xx3^(3) xx5^(5) .sqrt(2)`
D. `2^(3) xx3^(3) xx5^(2) .sqrt(2)`

Answer

To find the length of the diagonal of a square given its area, we can follow these steps:

### Step 1: Understand the relationship between area and side length
The area \( A \) of a square is given by the formula:
\[ A = a^2 \]
where \( a \) is the length of one side of the square.

### Step 2: Given area
We are given the area of the square as:
\[ A = 2^4 \times 3^6 \times 5^{10} \text{ sq. cm} \]

### Step 3: Find the side length
To find the side length \( a \), we take the square root of the area:
\[ a = \sqrt{A} = \sqrt{2^4 \times 3^6 \times 5^{10}} \]

Using the property of square roots:
\[ \sqrt{x \times y} = \sqrt{x} \times \sqrt{y} \]
we can simplify:
\[ a = \sqrt{2^4} \times \sqrt{3^6} \times \sqrt{5^{10}} \]

Calculating each square root:
\[ \sqrt{2^4} = 2^{4/2} = 2^2 = 4 \]
\[ \sqrt{3^6} = 3^{6/2} = 3^3 = 27 \]
\[ \sqrt{5^{10}} = 5^{10/2} = 5^5 = 3125 \]

Now, we can multiply these results to find \( a \):
\[ a = 4 \times 27 \times 3125 \]

### Step 4: Calculate the value of \( a \)
First, calculate \( 4 \times 27 \):
\[ 4 \times 27 = 108 \]

Now, multiply \( 108 \) by \( 3125 \):
\[ a = 108 \times 3125 \]

Calculating this gives:
\[ a = 337500 \]

### Step 5: Find the diagonal
The diagonal \( d \) of a square can be found using the formula:
\[ d = a \sqrt{2} \]

Substituting \( a \):
\[ d = 337500 \times \sqrt{2} \]

### Final Answer
Thus, the length of the diagonal in cm is:
\[ d = 337500 \sqrt{2} \text{ cm} \]

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