Final answer:
A fraction with equivalent quantities in the numerator and denominator simplifies to 1, and multiplying by 1 does not change the quantity's value. Algebra requires applying operations equally to maintain an equation's balance, and unit cancellation is a key part of dimensional analysis in simplifying units.
Step-by-step explanation:
When simplifying expressions involving absolute values or fractions, understanding that a fraction with the same quantity in the numerator and denominator equals 1 is crucial. For instance, a fraction like 1/1 simplifies to 1 because the quantities cancel each other out. This concept also applies when the quantities are expressed in different units but are equivalent, such as 100 cm and 1 m, which are the same length. Hence, a fraction like 100 cm / 1 m would also simplify to 1.
If we multiply any quantity by 1, the value of the quantity remains unchanged. In algebra, maintaining equality by performing the same operation on both sides is essential. When faced with an equation and we simplify or perform algebraic operations, we ensure the two sides remain balanced by applying changes uniformly.
Similarly, unit cancellation is a part of dimensional analysis. When units from the numerator and denominator cancel out during multiplication or division, the resulting units are those that have not been canceled. For example, in the unit conversion 1 m / 1000 mm, the units of millimeters (mm) will be canceled, leaving meters (m) as the desired unit in the result.