Problem 37 - College Algebra with Modeling and Visualization

Question

If possible, simplify the expression by hand. If you cannot, approximate the answer to the nearest hundredth. Variables represent any real number. $$ \sqrt[4]{81} $$

Step-by-Step Solution

Answer

The expression simplifies to 3.

Step 1: Understand What Simplification Means

Simplifying the expression \( \sqrt[4]{81} \) means finding a simpler equivalent expression, if possible. Specifically, we want to determine whether 81 has a fourth root that is a whole number.

Step 2: Consider Perfect Powers

Recognize that 81 might be a perfect power of 4. Recall that a number is a perfect fourth power if it can be written as another number raised to the fourth power.

Step 3: Find the Number Raised to the Fourth Power

Try some small integer values to see if any make 81 when raised to the fourth power. We check if \(3^4 = 3 \times 3 \times 3 \times 3 = 81\).

Step 4: Verify the Perfect Power Property

Since \(3^4 = 81\), we found that 81 is indeed a perfect fourth power, allowing us to simplify \(\sqrt[4]{81}\) to 3.

Key Concepts

Fourth Root

When you see the expression \(\sqrt[4]{81}\), it indicates you're looking for the fourth root of 81. The fourth root is a special type of root that essentially asks the question: "What number, when raised to the power of 4, becomes 81?" In simpler terms, it's the opposite operation of raising a number to the fourth power.
To find the fourth root, you need to determine if 81 is a result of some number multiplied by itself four times. In this case, since 3 multiplied by itself four times (\(3 \times 3 \times 3 \times 3\)) equals 81, the fourth root of 81 is 3.
Understanding fourth roots can deepen your grasp of how powers and roots interact, which is essential for solving many math problems efficiently.

Perfect Powers

Perfect powers are numbers that result from raising another number to an integer power like squares, cubes, or fourth powers. Recognizing perfect powers is crucial for simplifying roots.
In the case of \(81 = 3^4\), we see that 81 is a perfect fourth power because it's formed by raising 3 to the power of 4. This means that when simplifying \(\sqrt[4]{81}\), we can directly conclude that it simplifies to 3.
Notably, knowing common perfect powers can save you the hassle of guesswork, such as memorizing that 16 is a perfect fourth power of 2, or 81 as seen here. This applies generally to simplifying expressions involving roots, especially higher ones.

Real Numbers

Real numbers include all kinds of numbers you encounter like whole numbers, fractions, and decimals. They encompass both rational numbers (such as 1/2 or 2) and irrational numbers (like \(\sqrt{2}\) or \(\pi\)).
When dealing with expressions involving roots like \(\sqrt[4]{81}\), it's important to remember that variables represent real numbers, so the roots could potentially be any real number as well. However, in this example, we're focused on finding the whole number solution, which is a subset of real numbers.
A solid understanding of real numbers is valuable because it provides the foundation for more complex topics in mathematics. It ensures you're aware of all potential solutions when simplifying expressions or solving equations.