To add the fractions \( \frac{23}{144} + \frac{13}{1444} \), we need a common denominator. The least common multiple (LCM) of 144 and 1444 needs to be found.
First, let's factor the numbers:
- \( 144 = 12^2 = 2^4 \cdot 3^2 \)
- \( 1444 = 4 \cdot 361 = 4 \cdot 19^2 = 2^2 \cdot 19^2 \)
The LCM can be determined by taking the highest power of each prime that appears:
- For \( 2 \): the maximum is \( 2^4 \)
- For \( 3 \): the maximum is \( 3^2 \)
- For \( 19 \): the maximum is \( 19^2 \)
Thus, the LCM of 144 and 1444 is: \[ LCM(144, 1444) = 2^4 \cdot 3^2 \cdot 19^2 \]
Calculating it step-by-step: \[ 2^4 = 16,\quad 3^2 = 9,\quad 19^2 = 361 \] Now, multiply them together: \[ 16 \cdot 9 = 144 \] \[ 144 \cdot 361 = 51864 \]
Now we will convert each fraction to have a denominator of 51864.
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For \( \frac{23}{144} \): We convert it as follows: \[ \frac{23}{144} \cdot \frac{361}{361} = \frac{23 \cdot 361}{51864} = \frac{8293}{51864} \]
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For \( \frac{13}{1444} \): We convert it as follows: \[ \frac{13}{1444} \cdot \frac{36}{36} = \frac{13 \cdot 36}{51864} = \frac{468}{51864} \]
Now we can add the two fractions: \[ \frac{8293}{51864} + \frac{468}{51864} = \frac{8293 + 468}{51864} = \frac{8361}{51864} \]
Next, we simplify \( \frac{8361}{51864} \). We can find the greatest common divisor (GCD) to reduce the fraction:
Using the Euclidean algorithm or prime factorization, we get: \[ \text{GCD}(8361, 51864) = 9 \] So, we can divide both the numerator and denominator by 9: \[ \frac{8361 \div 9}{51864 \div 9} = \frac{929}{5760} \]
So the final simplified answer is:
- Numerator: 929
- Denominator: 5760