Problem 66 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 57-80: Simplify the expression, and rationalize the denominator when appropriate. $$ \sqrt[4]{81 r^{5} s^{8}} $$

Step-by-Step Solution

Answer

The simplified expression is \( 3 r s^{2} r^{1/4} \).

Step 1: Identify the Fourth Root

The expression given is \( \sqrt[4]{81 r^{5} s^{8}} \). To simplify it, we need to determine the fourth root of each component within the radical separately: \(81, r^{5}, \) and \(s^{8}\).

Step 2: Simplify the Constants

First, simplify the constant 81. Since \(81 = 3^4\), the fourth root of 81 is \(3\). Therefore, \( \sqrt[4]{81} = 3 \).

Step 3: Simplify the Variable \( r^{5} \) Using Exponent Rules

For \( r^{5} \), apply the exponent rule for roots: \( \sqrt[4]{r^5} = r^{5/4} = r^{1 + 1/4} = r \cdot r^{1/4} \). This means \( \sqrt[4]{r^5} = r \cdot r^{1/4} \).

Step 4: Simplify the Variable \( s^{8} \)

For \( s^{8} \), use the same concept: \( \sqrt[4]{s^8} = s^{8/4} = s^{2} \). This exact result is a perfect power of 4.

Step 5: Combine the Results

The simplified expression now combines each part: \( 3 \times r \times r^{1/4} \times s^{2} = 3 r s^{2} \times r^{1/4} \).

Step 6: Rationalize if Needed (Not Necessary Here)

We check if the denominator needs rationalizing; however, in this situation, there is no denominator in the result. So, this step confirms the simplified expression: \( 3 r r^{1/4} s^{2} \).

Key Concepts

Fourth Roots

When you see an expression like \( \sqrt[4]{81 r^{5} s^{8}} \), it's asking you to find the fourth root of each part of the expression. An important thing to remember about fourth roots is that they "undo" raising numbers and variables to the fourth power. For numbers, though, it's particularly straightforward: just ask yourself, "what number times itself four times gives me this number?" For example, with the number 81, since it's equal to \( 3^4 \), the fourth root is 3.

For variables with exponents, the process follows the same idea but requires the use of exponent rules, which we will discuss next. To find the fourth root of \( r^5 \) or \( s^8 \), you can apply these rules directly to simplify them into more manageable parts.

Exponent Rules

Exponent rules are super useful for manipulating expressions that have powers and roots. If you're dealing with fourth roots and an expression like \( r^5 \), you're going to use the rule that says you divide the exponent by the root. This turns \( \sqrt[4]{r^5} \) into \( r^{5/4} \). When simplifying, always try to break these down into understandable increments.

Another handy tip is memorizing that any variable raised to an exponent can be expressed as a multiplication of other variables, for instance, \( r^{5/4} = r^{1 + 1/4} = r \cdot r^{1/4} \). Breaking things down this way can make them neater and easier to work with. The imperativeness of exponent rules comes to the forefront especially when you handle expressions with roots or fractional exponents.

Rationalizing Denominators

Rationalizing the denominator means eliminating any roots in the bottom part of a fraction. However, in some expressions like \( 3 r s^2 \times r^{1/4} \), you don't have real denominators to rationalize, much less any root popping up from them.

In practice, if you had something like \( \frac{1}{\sqrt[4]{r^5}} \), you would want to get rid of the root when simplifying. This strategy can be a necessary step when you aim for a simplified expression that is also easy to compute or understand. Rationalizing generally leads to tidier results, making calculations simpler.

Algebraic Expressions

An algebraic expression like \( \sqrt[4]{81 r^5 s^8} \) involves numbers and variables, and simplifying it requires a solid grasp of algebra. The trick with these expressions is breaking them down into smaller distinct parts you can manage.

• The constant \(81 \) was broken down to \(3\) by taking the fourth root.
• The variable \(r^5\) was expressed as \( r \cdot r^{1/4} \)
• The variable \(s^8\) became \(s^2\).

These small steps are what help simplify the expression altogether into something like \( 3 r r^{1/4} s^2 \). Algebraic expressions like this can seem a bit tangled at first glance. But remember, they often just need to be broken down step-by-step using exponent rules and basic algebraic maneuvers.