Simplify: `((4)/(7)+(-8)/(9) +(-5)/(21) +(1)/(3))`.

Options

A. `(2)/(9)`
B. `(-1)/(9)`
C. `(-2)/(9)`
D. none of these

Answer

To simplify the expression \(\frac{4}{7} + \left(-\frac{8}{9}\right) + \left(-\frac{5}{21}\right) + \frac{1}{3}\), we will follow these steps:

### Step 1: Identify the denominators
The denominators in the expression are 7, 9, 21, and 3.

### Step 2: Find the Least Common Multiple (LCM)
We need to find the LCM of the denominators (7, 9, 21, 3).

- The prime factorization of each number is:
- \(7 = 7^1\)
- \(9 = 3^2\)
- \(21 = 3^1 \times 7^1\)
- \(3 = 3^1\)

The LCM will take the highest power of each prime:
- \(3^2\) from 9
- \(7^1\) from 7

Thus, the LCM is \(63\).

### Step 3: Convert each fraction to have the common denominator
Now we will convert each fraction to have a denominator of 63:

1. \(\frac{4}{7} = \frac{4 \times 9}{7 \times 9} = \frac{36}{63}\)
2. \(-\frac{8}{9} = -\frac{8 \times 7}{9 \times 7} = -\frac{56}{63}\)
3. \(-\frac{5}{21} = -\frac{5 \times 3}{21 \times 3} = -\frac{15}{63}\)
4. \(\frac{1}{3} = \frac{1 \times 21}{3 \times 21} = \frac{21}{63}\)

### Step 4: Combine the fractions
Now we can combine all the fractions:

\[ \frac{36}{63} - \frac{56}{63} - \frac{15}{63} + \frac{21}{63} \]

Combining the numerators:

\[ 36 - 56 - 15 + 21 = 36 - 56 = -20 \]
\[ -20 - 15 = -35 \]
\[ -35 + 21 = -14 \]

So we have:

\[ \frac{-14}{63} \]

### Step 5: Simplify the fraction
Now we simplify \(\frac{-14}{63}\):

The greatest common divisor (GCD) of 14 and 63 is 7. So we divide both the numerator and the denominator by 7:

\[ \frac{-14 \div 7}{63 \div 7} = \frac{-2}{9} \]

### Final Answer
The simplified form of the expression is:

\[ \frac{-2}{9} \]