Problem 31 - College Algebra

Question

Use the quotient rule to simplify the expressions in Exercises. Assume that \(x>0.\) $$\frac{\sqrt{200 x^{3}}}{\sqrt{10 x^{-1}}}$$

Step-by-Step Solution

Answer

The simplified expression of \( \frac{\sqrt{200 x^{3}}}{\sqrt{10 x^{-1}}}\) is \( 10 (x \sqrt{2x/10})\).

Step 1: Simplify square roots separately

Simplify numerator and denominator separately, first:For numerator: \(\sqrt{200 x^3} = \sqrt{(100*2)(x^2*x)}= 10x \sqrt{2x}\)For denominator: \(\sqrt{10 x^{-1}} = \sqrt{(10)(x^{-1})}= \sqrt{10}*x^{-1/2}\)

Step 2: Use Quotient Rule

Put the simplified numerator over the simplified denominator and apply the Quotient rule to simplify:\[\frac{10x \sqrt{2x}}{\sqrt{10}*x^{-1/2}} = 10x^{1+1/2} \sqrt{2x} * \sqrt{10}^{-1}= 10x^{3/2} \sqrt{2x} * \sqrt{10}^{-1}\].

Step 3: Simplify the Expression

\[10x^{3/2} \sqrt{2x} * \sqrt{10}^{-1}= 10x^{3/2} \sqrt{2x/10} = 10 (x \sqrt{2x/10})\].

Key Concepts

Simplifying Expressions

Simplifying mathematical expressions makes them easier to work with. It involves reducing the expression to its simplest form without changing its value. This can remind us of tidying a messy room — though the contents remain, they are arranged neatly. In our problem, we start by simplifying square roots in both the numerator and the denominator separately. The goal is to prepare each part of the expression for further reduction. Throughout this process, keeping track of coefficients and exponents ensures the expression is as compact and straightforward as possible. After simplification, things get clearer, making it easier to solve or understand the expression.

Square Roots

Square roots are a way to find a number which, when multiplied by itself, gives the original number. Simplifying square roots might seem tricky at first, but imagine peeling off layers to get to the core of an onion. You simply look for perfect squares, both numbers and variable expressions, within the square root. In our exercise, we dissect \(\sqrt{200 x^3}\) into its simplest form by recognizing \(200\) as \(100 \times 2\) and \(x^3\) as \(x^2 \times x\). Doing this lets us separate it as \(10x \sqrt{2x}\). Similarly, in the denominator, \(\sqrt{10 x^{-1}}\) breaks down to \(\sqrt{10} \times x^{-1/2}\). Reducing square roots to their simplest forms makes further calculation steps more manageable.

Exponents

Exponents represent how many times a number or variable is multiplied by itself. They are crucial in simplifying algebraic expressions. The trick with exponents is to remember the rules that guide operations like multiplication and division. For instance, when multiplying powers with the same base, you add the exponents; when dividing, you subtract. In our example, in using the Quotient Rule, we see how exponents behave when simplifying the expression \(\frac{10x^{3/2} \sqrt{2x}}{\sqrt{10}*x^{-1/2}}\). The results hinge on the careful handling of these exponents, where calculations like adding \(1\) and \(1/2\) deliver the precise simplification needed.

Rational Expressions

Rational expressions are like fractions in algebra, where the numerator and denominator are both polynomials. Simplifying them is akin to refining fractions by eliminating common factors. In this exercise, the role of rational expressions becomes apparent when applying the Quotient Rule. After independently simplifying the numerator and the denominator, putting them back together simplifies the expression as a whole. The techniques we apply simplify irrational numbers and variables, giving an expression that is more manageable and concise. Simplifying rational expressions allows further use in larger equations or complex functions, transforming them into forms that are straightforward to analyze or solve.