Problem 73 Write each quotient in lowest te... [FREE SOLUTION] (2024)

Short Answer

Factoring

Factoring is a key step in simplifying fractions, as seen in the exercise. It involves breaking down a number or an expression into its component parts or 'factors'—values that multiply together to produce the original number. For example, the problem involves the expression \((12 - 2\sqrt{10})/4\). To factor the numerator \(12 - 2\sqrt{10}\), you need to find a common factor in both terms. In this case, the common factor is 2 because 12 and -2 both divide evenly by 2. Factor out the 2 like so: \[12 - 2\sqrt{10} = 2(6 - \sqrt{10}).\] By doing this, you set up the fraction for further simplification. The new fraction becomes \[\frac{2(6 - \sqrt{10})}{4}.\] This makes it easier to cancel out common factors in the numerator and the denominator.

Square Roots

Square roots are expressions that represent the value which, when multiplied by itself, gives the original number. In the given exercise, one of the terms involves a square root: \(\sqrt{10}\). Square roots can often complicate fractions, but they can still be simplified. The term \(\sqrt{10}\) refers to the number that, when squared, equals 10. While \(\sqrt{10}\) is irrational and doesn't simplify neatly into a whole number, it still behaves like any other number in arithmetic operations. For example, in the expression \(6 - \sqrt{10}\), you treat \(\sqrt{10}\) as a single term. Simplifying expressions with square roots generally involves working on the non-radical terms separately, then rejoining them in the final step.

Common Factors

Common factors are numbers or expressions that divide into others without leaving a remainder. Identifying common factors simplifies both numerical and algebraic expressions. In the given problem, we were able to factor the numerator because both 12 and -2 share a common factor of 2. After factoring it out, the numerator looks like this: \[2(6 - \sqrt{10}).\] The denominator is 4, which also has 2 as a factor. By canceling out the common factor of 2 from the numerator and the denominator, you simplify the fraction: \[\frac{2(6 - \sqrt{10})}{4} = \frac{6 - \sqrt{10}}{2}.\] Now, the expression is simplified to its lowest terms. Always look for common factors when simplifying fractions to ensure you get the simplest form.