Problem 30 - College Algebra with Corequisite Support

Question

For the following exercises, simplify each expression. \(\sqrt[4]{16}\)

Step-by-Step Solution

Answer

The simplified expression is 2.

Step 1: Understanding the Problem

This exercise asks us to simplify the expression \(\sqrt[4]{16}\). Simplifying means finding a number which when raised to the 4th power gives 16.

Step 2: Recognizing the Base Expression

The expression \(\sqrt[4]{16}\) is the same as asking: what number to the power of 4 equals 16? In this case, we are looking for the fourth root of 16.

Step 3: Testing Possible Values

Recall that \(2^4 = 2 \times 2 \times 2 \times 2 = 16\). Therefore, 2 raised to the 4th power results in 16.

Step 4: Verification

To ensure correctness, verify that \((2)^4 = 16\). Indeed, \(2 \times 2 \times 2 \times 2 = 16\), confirming that 2 is the fourth root of 16.

Key Concepts

Fourth Root

The fourth root of a number is a value that, when raised to the fourth power, equals that original number. It acts as the inverse operation to raising a number to the fourth power. To simplify expressions involving the fourth root, you need to determine what number, when multiplied by itself four times, results in the given number inside the root.

• If you have a number, say 16, the expression \(\sqrt[4]{16}\) is asking you to find a number "x" such that \(x^4 = 16\).
• In our original problem, this number is 2, because \(2^4 = 2 \times 2 \times 2 \times 2 = 16\).

Finding the fourth root is quite similar to finding the square root, but instead of looking for two identical factors, you are looking for four. It's a useful technique in algebra to simplify expressions where the power involved is 4.

Exponents

Exponents are a shorthand way of expressing repeated multiplication of the same number by itself. In mathematical terms, an exponent shows how many times a number, known as the base, is multiplied by itself.

• For instance, \(2^4\) means 2 is used as a factor four times: \(2 \times 2 \times 2 \times 2\).
• The base here is 2, and the exponent is 4.

Exponents are pivotal in understanding roots like the fourth root. Recognizing the powers helps simplify roots by determining which numbers fit. When the exponent is 4, you're dealing with fouriterative multiplication, and the opposite is finding the fourth root.
For example, with \(\sqrt[4]{16}\), you determine which base number raised to the fourth power results in 16. Knowing how exponents and roots relate is essential in algebra, allowing you to solve complex problems by breaking them into simpler tasks.

Number Theory

Number theory is a branch of mathematics dealing with integers and the relationships between them. When working with expressions like \(\sqrt[4]{16}\), number theory helps you determine factors and powers that make computation easier.

• It involves concepts such as divisibility, prime numbers, and the fundamental theorem of arithmetic.
• Each number can be factored uniquely into a product of primes, essential in simplification and finding roots.

In our example, 16 can be expressed in terms of its prime factors: \(16 = 2^4\). Understanding these relationships helps simplify the problem to find that the fourth root of 16 is 2.
Number theory provides the foundational understanding needed to not only perform basic operations but also solve more advanced algebraic expressions. It helps illustrate the structure within numbers, guiding calculations, simplifications, and the derivation of roots.