Problem 26 - College Algebra

Question

Use the quotient rule to simplify the expressions in Exercises \(17-26 .\) Assume that \(x>0\) $$\frac{\sqrt{500 x^{3}}}{\sqrt{10 x^{-1}}}$$

Step-by-Step Solution

Answer

The simplified expression of \(\frac{\sqrt{500 x^{3}}}{\sqrt{10 x^{-1}}}\) is \(2\sqrt{5}x^{2}\).

Step 1: Simplifying the square roots

First, each of the two square root expressions in the fraction is simplified. Start with \(\sqrt{500 x^{3}}\). Using properties of square roots, this expression becomes \(10\sqrt{5}x^{\frac{3}{2}}\). Now look at \(\sqrt{10 x^{-1}}\), this expression becomes \(\sqrt{10}x^{-\frac{1}{2}}\). Now the original expression turns into \(\frac{10\sqrt{5}x^{\frac{3}{2}}}{\sqrt{10}x^{-\frac{1}{2}}}\).

Step 2: Applying the quotient rule of exponents

When dividing terms with the same base, subtract the exponent in the denominator from the exponent in the numerator. This gives \(\frac{10\sqrt{5}x^{\frac{3}{2}}}{\sqrt{10}x^{-\frac{1}{2}}}=10\sqrt{5}x^{\frac{3}{2}-\left(-\frac{1}{2}\right)}\div\sqrt{10}\). Simplify the formula and it becomes \(\frac{10\sqrt{5}x^{2}}{\sqrt{10}}\).

Step 3: Simplifying the final expression

The last step is to simplify the fraction by cancelling similar terms. The expression \(\frac{10\sqrt{5}x^{2}}{\sqrt{10}}\) simplifies to \(2\sqrt{5}x^{2}\). It is not necessary to normalize the square root part further for this exercise.

Key Concepts

Understanding the Quotient Rule

When working with square roots and fractions, the quotient rule is a fundamental concept to grasp. It states that when you have a fraction under a square root, you can take the square root of the numerator and the denominator separately. This allows for simplification of complex expressions.

For example, with the expression \(\frac{\sqrt{500 x^{3}}}{\sqrt{10 x^{-1}}}\), the task is to simplify each square root individually before applying the quotient rule. The goal is to make the expression as concise as possible without changing its value. Next, let's dive into how we can further use the quotient rule in combination with the properties of square roots and exponents to simplify expressions.

Properties of Square Roots

Square roots have unique properties that allow us to manipulate and simplify expressions. One key property is that the square root of a product \(\sqrt{a \cdot b}\) is equal to the product of the square roots \(\sqrt{a} \cdot \sqrt{b}\). Likewise, the square root of a quotient \(\sqrt{\frac{a}{b}}\) is the quotient of the square roots \(\frac{\sqrt{a}}{\sqrt{b}}\). These properties are essential when it comes to simplifying the square roots of numbers and variables with exponents.

For our exercise, \(\sqrt{500 x^{3}}\) is broken down into its prime factors and variables, and similarly for \(\sqrt{10 x^{-1}}\). This step-by-step approach allows us to handle each part of the expression in a more manageable way, setting the stage for further simplification using the quotient rule and the rules of exponents.

Simplifying with Exponents

Exponents play a significant role in simplifying mathematical expressions, especially when combined with square roots. An essential rule to remember is the quotient rule for exponents, which states that when dividing like bases, you subtract the exponents \(x^{a}/x^{b} = x^{a-b}\). This rule simplifies the process of combining and reducing terms in fractional exponents.

In the exercise, we use this rule to combine \(x^{\frac{3}{2}}\) and \(x^{-\frac{1}{2}}\). Subtracting the exponents gives us a simplified exponent of \(x^{2}\), streamlining the original expression to \(2\sqrt{5}x^{2}\). Understanding this concept is powerful for simplifying a range of expressions involving exponents, and it is particularly useful in algebra and higher mathematics.