Problem 30 - College Algebra

Question

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

Step-by-Step Solution

Answer

\( \sqrt[4]{16} = 2 \)

Step 1: Determine the Fourth Root

To simplify the expression \( \sqrt[4]{16} \), we need to find a number that, when raised to the fourth power, equals 16. This is known as finding the fourth root of 16.

Step 2: Find the Unknown Value

Consider whether there is an integer that satisfies this condition. Since \(2^4 = 2 \times 2 \times 2 \times 2 = 16 \), we see that 2 is the number we are looking for.

Step 3: Simplify the Expression

Substitute the result from Step 2 into the expression. Thus, \( \sqrt[4]{16} = 2 \).

Key Concepts

Fourth Root

The fourth root of a number is similar to finding the more commonly known square root, but instead of looking for a number which multiplies by itself twice to obtain the original number, we look for a number that does so four times. If we have the expression \( \sqrt[4]{16} \), we are essentially asking: "What number raised to the power of four gives us 16?" To find the fourth root:

• Identify a potential number and multiply it by itself four times.
• Check if the result equals the original number, 16 in this case.
• If it fits, you have found your fourth root!

In this exercise, we found that when 2 is multiplied by itself four times \((2 \times 2 \times 2 \times 2 = 16)\), it equals 16. Thus, \( \sqrt[4]{16} \) simplifies to 2.

Integer Solutions

An integer solution refers to a whole number result from a mathematical problem or equation, which includes positive numbers, negative numbers, and zero. In this case with \( \sqrt[4]{16} \), the problem requires finding an integer that can be raised to the fourth power to equal 16.The process of finding integer solutions can involve:

• Checking potential integer candidates by progressively raising them to the power specified – starting from lower numbers for practicality.
• Observing if any of these trials result in the target number.

For \( \sqrt[4]{16} \), we tested integer values and found that 2 was the correct solution because \(2^4 = 16\). This shows that 2 is the fourth root of 16, making it the integer solution to this problem.

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The base is the number that we are going to multiply, and the exponent indicates how many times the base is multiplied by itself. For example, in the expression \(2^4\), 2 is the base and 4 is the exponent, which means \(2 \times 2 \times 2 \times 2\).Understanding Exponents:

• The exponent tells you how many times to use the base as a factor.
• An exponent of 4, like in \(2^4\), means you multiply the base (2) by itself, four times.

In the context of our problem, exponentiation plays a crucial role. We were looking for a base that, when raised to the power of 4, matches the number within the radical, 16. Through exponentiation, we determined that \(2^4 = 16\), which confirmed our solution. Exponentiation not only helps in calculations but also in understanding the fundamental structure of numbers and their powers.