Simplify the following expressions :
`9/7 xx 21/6 xx 5/8 xx 3 5/6`

Answer

To simplify the expression \( \frac{9}{7} \times \frac{21}{6} \times \frac{5}{8} \times 3 \frac{5}{6} \), we will follow these steps:

### Step 1: Convert the mixed number to an improper fraction
The mixed number \( 3 \frac{5}{6} \) can be converted to an improper fraction:
\[ 3 \frac{5}{6} = \frac{3 \times 6 + 5}{6} = \frac{18 + 5}{6} = \frac{23}{6} \]

### Step 2: Rewrite the expression
Now, we can rewrite the expression with the improper fraction:
\[ \frac{9}{7} \times \frac{21}{6} \times \frac{5}{8} \times \frac{23}{6} \]

### Step 3: Multiply the fractions
We can multiply the numerators and the denominators:
\[ \text{Numerator: } 9 \times 21 \times 5 \times 23 \]
\[ \text{Denominator: } 7 \times 6 \times 8 \times 6 \]

### Step 4: Simplify the fractions
Before multiplying, we can simplify the fractions by canceling common factors:
- \( 21 = 3 \times 7 \) (so \( 7 \) in the denominator cancels with \( 21 \))
- \( 9 = 3 \times 3 \) (so \( 3 \) in the numerator cancels with \( 6 \) in the denominator)
- \( 6 \) in the denominator cancels with \( 6 \) in the numerator from \( 21 \)

After canceling, we have:
\[ \frac{3 \times 5 \times 23}{2 \times 8} \]

### Step 5: Calculate the remaining values
Now we can compute the numerator and denominator:
- Numerator: \( 3 \times 5 \times 23 = 345 \)
- Denominator: \( 2 \times 8 = 16 \)

So we have:
\[ \frac{345}{16} \]

### Step 6: Convert to mixed number
To convert \( \frac{345}{16} \) to a mixed number:
- Divide \( 345 \) by \( 16 \):
\[ 345 \div 16 = 21 \quad \text{(whole number part)} \]
- Remainder: \( 345 - (21 \times 16) = 9 \)

Thus, the mixed number is:
\[ 21 \frac{9}{16} \]

### Final Answer
The simplified expression is:
\[ 21 \frac{9}{16} \]

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