Question

Add. Simplify the answer to lowest terms before entering the numerator and denominator in their boxes.

23/144
+
13/1444

Answer 1

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.

1. For \( \frac{23}{144} \):
We convert it as follows:
\[
\frac{23}{144} \cdot \frac{361}{361} = \frac{23 \cdot 361}{51864} = \frac{8293}{51864}
\]

2. 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