Final answer:
The expression a/b = c/d can be simplified by cross-multiplying to get ad = bc and simplifying further as needed. For exponentials, divide numerators and subtract exponents with the same basis. Dimensional analysis helps in simplifying algebra by ensuring units cancel out correctly.
Step-by-step explanation:
To find and simplify the expression given that a/b = c/d, we can approach this problem by cross-multiplication or by using properties of proportions. Essentially, if two fractions are equal, their cross products are also equal, meaning ad = bc. Once we have this relationship, we can simplify by factoring, reducing common factors, or applying some algebraic manipulations to solve for a particular variable, if that's required.
Let's take another scenario where we have exponentials. For division of exponentials, you would divide the digit term of the numerator by the digit term of the denominator and subtract the exponents if the bases of the exponentials are the same. This follows the property that am / an = am-n.
If the problem requires solving for a variable such as y, we would manipulate the equation accordingly. For example, if we have y = (a/b) * (c/d), and we want to isolate y, we can simply use multiplication and division to manipulate the equation. If y needs to be expressed in terms of another variable, we'd perform further algebraic steps.
When we multiply or divide fractions in units, it's important to eliminate terms wherever possible to simplify the algebra. This also includes checking that units cancel appropriately, leaving us with the desired units in the result. This is an application of dimensional analysis.