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When working with fractions, simplifying means rewriting the fraction in its simplest form. This makes it easier to understand and compare. To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator (top number) and the denominator (bottom number). The GCD is the largest number that divides both numbers without a remainder.

For example, in the fraction \(\frac{-32}{42}\), both -32 and 42 are divisible by 2, their GCD. By dividing both the numerator and the denominator by 2, we get: \(\frac{-32 梅 2}{42 梅 2} = \frac{-16}{21}\).

This gives us the simplest form of \(\frac{-32}{42}\), which is \(\frac{-16}{21}\). Remember, always try to simplify your fractions after solving any problem, as it provides the clearest answer.

A reciprocal of a fraction is found by flipping the numerator and denominator. Essentially, you 鈥渢urn the fraction upside down.鈥� For a fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\).

Reciprocals are very useful when dividing fractions. Instead of directly dividing, you multiply by the reciprocal of the divisor. For example, in \(\frac{1}{2} 梅 \frac{3}{4} \), we don't divide by \(\frac{3}{4}\), but instead multiply by its reciprocal: \(\frac{1}{2} \times \frac{4}{3}\).

So, in our problem, the division \(\frac{1}{2} \/ \left(鈭抃frac{3}{4}\right) \/ \frac{7}{8}\) transforms into: \(\frac{1}{2} \times 鈭抃frac{4}{3} \times \frac{8}{7}\).

By finding reciprocals, fractions become multipliable, making the solution simpler and more approachable.

Multiplying fractions is straightforward. You only need to multiply the numerators together and then the denominators together.

Here's a step-by-step guide:

• Multiply the numerators: 1 \(\times\) -4 \(\times\) 8 = -32.
• Multiply the denominators: 2 \(\times\) 3 \(\times\) 7 = 42.

The result of multiplying \(\frac{1}{2} \times 鈭抃frac{4}{3} \times \frac{8}{7}\) is \(\frac{-32}{42}\).

After multiplying, you should look to simplify the fraction, as discussed earlier.

Knowing how to multiply fractions is also essential for other mathematical operations where fractions are involved.