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Fraction simplification helps to make fractions easier to work with by reducing them to their simplest form. When you simplify a fraction, you divide the numerator and the denominator by their greatest common divisor (GCD). For instance, if you have a fraction \(\frac{6}{8}\), you find that both 6 and 8 can be divided by 2. So, \(\frac{6}{8} \rightarrow \frac{6 \div 2}{8 \div 2} = \frac{3}{4}\). Simplified fractions are useful because they make both multiplication and addition of fractions easier.

Denominator adjustment is the process used to rewrite a fraction with a different denominator. This concept is essential when you need to compare, add, or subtract fractions. To adjust the denominator, a multiplicative factor is used. For example, in the exercise, we want to rewrite \(\frac{3}{-7}\) with a new denominator of 70. We look for a number that, when multiplied by the original denominator, \(-7\), gives 70. Solving for the multiplicative factor involves dividing the new denominator by the old one: \(\frac{70}{-7} = -10\). Then, both the numerator and the denominator of the original fraction are multiplied by this factor to get the new fraction.

A multiplicative factor is the number you use to scale the numerator and the denominator of a fraction. To find it, often involves a simple division. In our exercise, to change the fraction \(\frac{3}{-7}\) to have a denominator of 70, we determine the factor by solving \(-7x = 70\). Dividing by \(-7\) gives \(-10\), which is our multiplicative factor. By multiplying the numerator and the denominator of \(\frac{3}{-7}\) by \(-10\), we ensure the fraction remains equivalent while adjusting its form: \(\frac{3 \times (-10)}{-7 \times (-10)} = \frac{-30}{70}\). Using consistent multiplicative factors helps maintain the value of fractions while allowing flexibility in their representation.