Simplify: (i) `(-4/7) xx (21/20)`, (ii) `(-13)/9 xx (-21)/(-39)`

Answer

Let's simplify the given expressions step by step.

### Part (i): Simplify \((-4/7) \times (21/20)\)

**Step 1:** Write the expression.
\[ \frac{-4}{7} \times \frac{21}{20} \]

**Step 2:** Factor the numbers to simplify.
- The numerator of the first fraction is \(-4\) and the numerator of the second fraction is \(21\).
- The denominator of the first fraction is \(7\) and the denominator of the second fraction is \(20\).

**Step 3:** Simplify the fractions by finding common factors.
- \(21\) can be divided by \(7\) (since \(7 \times 3 = 21\)).
- \(-4\) can be divided by \(4\) (since \(4 \times 5 = 20\)).

So we can rewrite:
\[ \frac{-4}{7} \times \frac{21}{20} = \frac{-4 \div 4}{7 \div 7} \times \frac{21 \div 7}{20 \div 4} = \frac{-1}{1} \times \frac{3}{5} \]

**Step 4:** Multiply the simplified fractions.
\[ \frac{-1 \times 3}{1 \times 5} = \frac{-3}{5} \]

**Final Answer for Part (i):**
\[ \frac{-3}{5} \]

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### Part (ii): Simplify \(\frac{-13}{9} \times \frac{-21}{-39}\)

**Step 1:** Write the expression.
\[ \frac{-13}{9} \times \frac{-21}{-39} \]

**Step 2:** Simplify the fractions.
- The numerator of the first fraction is \(-13\) and the numerator of the second fraction is \(-21\).
- The denominator of the first fraction is \(9\) and the denominator of the second fraction is \(-39\).

**Step 3:** Cancel out the negative signs.
The negative signs in both fractions cancel each other out:
\[ \frac{-13}{9} \times \frac{-21}{-39} = \frac{13}{9} \times \frac{21}{39} \]

**Step 4:** Simplify \(\frac{21}{39}\).
- \(21\) can be divided by \(3\) (since \(3 \times 7 = 21\)).
- \(39\) can also be divided by \(3\) (since \(3 \times 13 = 39\)).

So we can rewrite:
\[ \frac{21}{39} = \frac{21 \div 3}{39 \div 3} = \frac{7}{13} \]

**Step 5:** Now multiply the simplified fractions.
\[ \frac{13}{9} \times \frac{7}{13} = \frac{13 \times 7}{9 \times 13} \]

The \(13\) in the numerator and denominator cancels out:
\[ = \frac{7}{9} \]

**Final Answer for Part (ii):**
\[ \frac{7}{9} \]

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