Problem 73 - Algebra 2

Question

Simplify each radical expression. Use absolute value bars where they are needed. $$ \sqrt[4]{64 m^{8} n^{4}} $$

Step-by-Step Solution

Answer

The simplified form of \( \sqrt[4]{64 m^{8} n^{4}} \) is \(2m^{2}|n|.\)

Step 1: Break Down Radical into Potential Components

Firstly, break down the components of the radical: \( \sqrt[4]{64}\), \( \sqrt[4]{m^{8}}\), and \( \sqrt[4]{n^{4}}\), which are the quartic roots of 64, \(m^{8}\), and \(n^{4}\) respectively.

Step 2: Simplify Each Component

Secondly, simplify each component. The quartic root of 64 is 2, the quartic root of \(m^{8}\) is \(m^{2}\), and the quartic root of \(n^{4}\) is \(|n|\). Note that \(|n|\) is used even though the quartic root of \(n^{4}\) is also \(n\), because \(n\) could be negative and the radical of any number should be nonnegative.

Step 3: Combine The Simplified Components

Thirdly, combine the simplified components to get the simplified form of the original expression, which results in \(2m^{2}|n|.\)

Key Concepts

Quartic Root

The quartic root is essentially the fourth root of a number. When we talk about the quartic root, we are looking for a number that, when multiplied by itself four times, gives the original number.
For example, the quartic root of 64 is 2 because when you multiply 2 by itself four times (2 \( \times \) 2 \( \times \) 2 \( \times \) 2), you get 64.

• This can be expressed mathematically as \( \sqrt[4]{64} = 2 \).

Understanding quartic roots can be crucial in algebra, especially when simplifying radical expressions like \( \sqrt[4]{64 m^{8} n^{4}} \). By breaking each part down into quartic roots, calculations become much simpler as you'll see in the simplification process.

Absolute Value

The absolute value of a number is its distance from zero on the number line, without regard to direction. It is always a positive number or zero, as it describes magnitude but not direction.
In mathematical expressions, absolute values are denoted with bars, e.g., |n|.

• This is important in radical simplification because, when taking roots, the result should be nonnegative.

For instance, when simplifying \( \sqrt[4]{n^4} \), the result is written as |n| because whether the original n was positive or negative, the result in absolute terms is the same. Therefore, absolute values ensure that the simplified expression reflects the correct mathematical principle of nonnegative roots.

Exponents

Exponents are used in mathematics to denote repeated multiplication of a number by itself. When taking the quartic root, you are essentially reversing the action of an exponent of 4.
For example, when you have \( m^8 \), it represents \( m \) multiplied by itself 8 times.

• In simplification, taking \( \sqrt[4]{m^8} \) involves dividing the exponent by 4, resulting in \( m^{8/4} \) or \( m^2 \).

This simplification rule is generally applied whenever you are determining the root of a number raised to an exponent. It makes handling polynomial expressions more manageable and aids in reducing complex expressions to simpler, more understandable forms.

Negative Numbers

Negative numbers are numbers less than zero, and they can sometimes complicate the process of simplification. This is because certain mathematical operations, like taking even roots, require nonnegative results.
For example, the quartic root and square root of a number both imply a nonnegative result.

• This is why absolute values might be used when simplifying expressions involving negative numbers, as seen in \( \sqrt[4]{n^4} = |n| \).

Being mindful of negative numbers and their properties ensures that simplifications are mathematically correct and logically consistent. It highlights the need to consider both the magnitude and sign of any number under even roots.