The number of digits in the product of `5^(72).8^(27)` is :

Options

A. a.77
B. b.75
C. c.99
D. d.none of (a),(b),(c)

Answer

To find the number of digits in the product of \(5^{72} \cdot 8^{27}\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression \(5^{72} \cdot 8^{27}\). We know that \(8\) can be expressed as \(2^3\). Therefore, we can rewrite \(8^{27}\) as:
\[ 8^{27} = (2^3)^{27} = 2^{81} \]
So, the expression becomes:
\[ 5^{72} \cdot 2^{81} \]

### Step 2: Combine the powers
Next, we can combine the powers of \(5\) and \(2\). We can factor out \(2^{72}\) from \(2^{81}\):
\[ 5^{72} \cdot 2^{81} = 5^{72} \cdot 2^{72} \cdot 2^{9} = (5 \cdot 2)^{72} \cdot 2^{9} \]
Since \(5 \cdot 2 = 10\), we can rewrite the expression as:
\[ 10^{72} \cdot 2^{9} \]

### Step 3: Calculate \(2^{9}\)
Now, we need to calculate \(2^{9}\):
\[ 2^{9} = 512 \]
Thus, we can rewrite our expression as:
\[ 10^{72} \cdot 512 \]

### Step 4: Determine the number of digits
To find the number of digits in a number, we can use the formula:
\[ \text{Number of digits} = \lfloor \log_{10}(\text{number}) \rfloor + 1 \]
Now, we need to find the number of digits in \(10^{72} \cdot 512\):
\[ \log_{10}(10^{72} \cdot 512) = \log_{10}(10^{72}) + \log_{10}(512) \]
Calculating these:
\[ \log_{10}(10^{72}) = 72 \]
And we need to find \(\log_{10}(512)\). Since \(512 = 2^9\), we can calculate:
\[ \log_{10}(512) = \log_{10}(2^9) = 9 \cdot \log_{10}(2) \]
Using \(\log_{10}(2) \approx 0.301\):
\[ \log_{10}(512) \approx 9 \cdot 0.301 = 2.709 \]

### Step 5: Combine the logarithms
Now we can combine the logarithms:
\[ \log_{10}(10^{72} \cdot 512) \approx 72 + 2.709 = 74.709 \]

### Step 6: Calculate the number of digits
Finally, we can find the number of digits:
\[ \text{Number of digits} = \lfloor 74.709 \rfloor + 1 = 74 + 1 = 75 \]

Thus, the number of digits in the product \(5^{72} \cdot 8^{27}\) is **75**.

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