how to find the number of digits in the product of a number raised to a power and with another number raised to a power

To find the number of digits in the product of two numbers, each raised to a power, you can follow these steps:

### General Scenario:
Let’s say you have two numbers, \( a \) and \( b \), raised to powers \( m \) and \( n \), respectively, such that you want to calculate the number of digits in:

\[
N = (a^m) \times (b^n)
\]

### Step-by-Step Process:

1. **Combine the Exponents**:
   First, combine the two terms into a single term:
   \[
   N = a^m \times b^n = (a^m \cdot b^n) = (a^m) \times (b^n)
   \]

2. **Take the Logarithm (Base 10)**:
   To find the number of digits, take the logarithm base 10 of the number \( N \):
   \[
   \log_{10}(N) = \log_{10}(a^m \cdot b^n)
   \]
   Using the property of logarithms:
   \[
   \log_{10}(N) = \log_{10}(a^m) + \log_{10}(b^n)
   \]
   Which simplifies to:
   \[
   \log_{10}(N) = m \cdot \log_{10}(a) + n \cdot \log_{10}(b)
   \]

3. **Calculate the Number of Digits**:
   The number of digits in a number \( N \) is given by the integer part of the logarithm of the number plus 1:
   \[
   \text{Number of digits in } N = \lfloor \log_{10}(N) \rfloor + 1
   \]
   Where \( \lfloor x \rfloor \) represents the floor function, which gives the largest integer less than or equal to \( x \).

### Example:

Let’s say you want to find the number of digits in \( 3^4 \times 5^3 \):

1. \( a = 3 \), \( m = 4 \)
2. \( b = 5 \), \( n = 3 \)
3. Take the logarithm:
   \[
   \log_{10}(N) = 4 \cdot \log_{10}(3) + 3 \cdot \log_{10}(5)
   \]
   Using approximations:
   \[
   \log_{10}(3) \approx 0.4771 \quad \text{and} \quad \log_{10}(5) \approx 0.69897
   \]
   So:
   \[
   \log_{10}(N) = 4 \cdot 0.4771 + 3 \cdot 0.69897 = 1.9084 + 2.09691 = 4.00531
   \]
4. Number of digits:
   \[
   \lfloor 4.00531 \rfloor + 1 = 4 + 1 = 5
   \]

So the number \( 3^4 \times 5^3 \) has **5 digits**.

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Learn how to determine the number of digits in the product of two numbers raised to powers using exponents and logarithms. This guide provides a step-by-step process with an example, combining logarithmic properties to simplify the calculation. Ideal for students in Grades 9-12.

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