Problem 29 - Algebra and Trigonometry

Question

Use the quotient rule to simplify the expressions Assume that \(x>0\). $$ \frac{\sqrt{150 x^{4}}}{\sqrt{3 x}} $$

Step-by-Step Solution

Answer

The simplified expression is \(10x^{3/2} \sqrt{5}\).

Step 1: Simplify the numerator

First, simplify \(\sqrt{150 x^{4}}\). The square root of \(150\) is \(10\sqrt{15}\) (since \(150 = 10^2 * 15\)), and the square root of \(x^4\) is \(x^2\) (since the square and the square root cancel out). Combining both, we get that \(\sqrt{150 x^{4}} = 10x^2 \sqrt{15}\).

Step 2: Simplify the denominator

Next, simplify \(\sqrt{3x}\). The square root of \(3x\) is \(\sqrt{3} \sqrt{x}\). However, since \(x>0\), \(\sqrt{x}\) is real and positive, which we will denote by \(x^{1/2}\). Therefore, \(\sqrt{3x} = x^{1/2} \sqrt{3}\).

Step 3: Simplify the whole expression

Now, divide the result from step 1 by the result from step 2, according to the quotient rule. Thus, \(\frac{10x^2 \sqrt{15}}{x^{1/2} \sqrt{3}} = \frac{10x^2}{x^{1/2}} * \frac{\sqrt{15}}{\sqrt{3}}\). Simplify \(\frac{10x^2}{x^{1/2}}\) to \(10x^{3/2}\) and \(\frac{\sqrt{15}}{\sqrt{3}}\) to \(\sqrt{5}\), to get the final simplified expression: \(10x^{3/2} \sqrt{5}\).

Key Concepts

Simplifying Expressions

Simplifying expressions is a key skill in algebra, making complex problems more manageable by reducing them to simpler forms. This process involves using different mathematical rules and properties to rewrite expressions in a clearer or more useful form. We basically work to make the expression less complex without changing its value.
A common approach to simplifying expressions is by combining like terms, where we put together terms that have the same variables raised to the same powers. Also, we can cancel out common factors in fractions when simplifying ratios or fractions. For example, in our problem, the quotient rule is used to divide terms, reducing the complexity of expressions by factoring like terms.
It's important to remember that simplifying expressions isn't just about making them shorter; it’s about making them easier to use in calculations. This simplification helps in solving equations or inequalities more easily. To successfully obtain a simplified form, you need to be familiar with various algebraic rules and how they apply in specific scenarios.

Square Roots

Square roots are the most common type of root, and they appear frequently when simplifying expressions. The square root of a number is a value that, when multiplied by itself, gives the original number.
The concept of square roots is crucial, especially when dealing with exponents and radicals, as seen in our original exercise.

• The symbol \( \sqrt{} \) represents the square root.
• Expressions like \( \sqrt{150} \) are simplified by factoring the number under the square root, which helps identify perfect squares.

In the exercise, we simplified \( \sqrt{150x^4} \) by breaking 150 into \(10^2\) and 15, revealing the perfect square that helps simplify further. Learning how square roots behave with products and factors is a handy tool in making sense of radical expressions.

Rational Exponents

Rational exponents, often known as fractional exponents, express roots and powers compactly. Instead of using radical signs, we express roots as fractions in exponents, which makes it easier to manipulate and combine terms.
Here’s how it works:

• A square root such as \(\sqrt{x}\) can be rewritten using rational exponents as \(x^{1/2}\).
• More generally, \(\sqrt[n]{x}\) is rewritten as \(x^{1/n}\).

With rational exponents, we can apply rules of exponents much more effectively. For instance, multiplying terms with the same base involves adding their exponents, a rule that's easier to apply when dealing with fractions rather than radicals. In our solution, rational exponents helped us simplify the denominator term \(\sqrt{3x}\), making it easier to divide by \(x^{1/2}\).

Algebraic Expressions

Algebraic expressions are the building blocks of algebra that combine numbers and variables to represent mathematical ideas. They can range from a single term to a combination of multiple terms linked by mathematical operators.
Understanding the structure of algebraic expressions is essential because they are central to formulating equations and functions.

• An algebraic expression like \(10x^2 \sqrt{15}\) consists of coefficients (10), variables (\(x\)), and radicals (\(\sqrt{15}\)).
• Complex expressions can be made manageable by breaking them into parts and simplifying step by step.

In our exercise, the expression was made simpler by recognizing factors and translating roots into exponents. This highlights the importance of examining each part of an expression, applying relevant algebraic rules, and carefully rearranging terms to achieve a simplified form. Whether in academics or real-world applications, mastery over interpreting and solving algebraic expressions is invaluable.