Problem 29 - Algebra 2 and Trigonometry

Question

In \(3-29\) write each quotient in simplest form. Variables in the radicand with an even index are non-negative. Variables occurring in the denominator of a fraction are non-zero. $$ \frac{\sqrt{64 x^{4}}+\sqrt[4]{40 x^{6}}}{\sqrt[4]{x^{6}}} $$

Step-by-Step Solution

Answer

The simplest form is \( 8x^{0.5} + \sqrt[4]{40} \).

Step 1: Simplify the Numerator

We have two terms in the numerator, \( \sqrt{64 x^{4}} \) and \( \sqrt[4]{40 x^{6}} \). Let's simplify them one by one.For \( \sqrt{64 x^{4}} \):\[ \sqrt{64 x^{4}} = \sqrt{(8x^{2})^{2}} = 8x^{2} \]For \( \sqrt[4]{40 x^{6}} \):\[ \sqrt[4]{40 x^{6}} = \sqrt[4]{(40)x^{4}x^{2}} = x^{1.5} \times \sqrt[4]{40} \] (we leave \( \sqrt[4]{40} \) as an expression since it cannot be simplified to an integer).

Step 2: Simplify the Denominator

The denominator is \( \sqrt[4]{x^{6}} \).To simplify it:\[ \sqrt[4]{x^{6}} = (x^{6})^{1/4} = x^{1.5} \]

Step 3: Replace Simplified Expressions

Substitute the simplified expressions back into the original quotient:\[ \frac{8x^{2} + x^{1.5} \times \sqrt[4]{40}}{x^{1.5}} \]

Step 4: Divide Each Term by the Denominator

Divide each term in the numerator by \( x^{1.5} \):\[ \frac{8x^{2}}{x^{1.5}} + \frac{x^{1.5} \times \sqrt[4]{40}}{x^{1.5}} \]This simplifies to:\[ 8x^{0.5} + \sqrt[4]{40} \]

Step 5: Simplified Expression

So, the expression \( \frac{8x^{0.5} + \sqrt[4]{40}} \) is the simplest form of the given quotient.

Key Concepts

Understanding Radicands

In radical expressions like \( \sqrt{64 x^{4}} \) and \( \sqrt[4]{40 x^{6}} \), the term inside the radical is known as the radicand. The radicand can consist of numbers, variables, or both. When simplifying radicals, focus on the radicand to determine if it can be reduced. For instance:

• \( \sqrt{64 x^{4}} \) can be rewritten as \( \sqrt{(8x^{2})^{2}} \). Here, both 64 and \( x^4 \) are perfect squares, making it easy to simplify to \( 8x^{2} \).
• \( \sqrt[4]{40 x^{6}} \) is a bit more complex because 40 is not a perfect fourth power. We separate out the perfect fourth powers in the radicand: \( x^4 \) and \( x^2 \) allow for simplification to \( x^{1.5} \times \sqrt[4]{40} \).

In general, focus on breaking down the radicand into smaller segments that can be simplified based on their power or numerical value.

Mastering Fraction Simplification

Simplifying fractions in mathematical expressions involves reducing fractions to their simplest form. This requires dividing each term by the greatest common factor, especially when dealing with radicals. The process often involves distributing the terms and dealing with variables carefully:

• Consider the fraction \( \frac{8x^{2} + x^{1.5} \times \sqrt[4]{40}}{x^{1.5}} \). Here, we simplify by dividing both terms in the numerator by \( x^{1.5} \).
• This yields: \( \frac{8x^{2}}{x^{1.5}} + \frac{x^{1.5} \times \sqrt[4]{40}}{x^{1.5}} \).
• The fraction simplifies to \( 8x^{0.5} + \sqrt[4]{40} \) after canceling out like terms.

Remember to start by simplifying each part of the fraction separately, and ensure you divide just like terms effectively to avoid errors.

Exploring the Properties of Exponents

Exponents play a key role in simplifying algebraic expressions, especially when working with radicals and fraction exponents. Understanding and applying the properties of exponents helps in reducing complex expressions effectively. Here are some fundamentals:

• The power of a power property: \( (x^m)^n = x^{m \times n} \). When simplifying \( \sqrt{64 x^4} = 8x^2 \), you leverage the fact that \( x^4 \) can be reduced using square root properties.


• The product rule: \( x^m \times x^n = x^{m+n} \). This property is used in combining like terms with exponents such as when simplifying \( 8x^2 \) and \( x^{1.5} \).


• Rational exponents: \( x^{1/n} = \sqrt[n]{x} \). Converting between radical forms and rational exponents assists in simplification, as seen with \( \sqrt[4]{x^6} = x^{1.5} \).

Applying these properties lets you navigate and simplify expressions more effectively, leading to clean and simplified outcomes.