Problem 34 - Fundamentals of Mathematics

Question

Find each value. Check each result with a calculator. \(\sqrt[4]{16}-1+5^{2}\)

Step-by-Step Solution

Answer

The value of the expression is 26.

Step 1: Calculate the Fourth Root of 16

To begin, calculate the fourth root of 16, which is denoted by \(\sqrt[4]{16}\). The fourth root of a number is a value that, when multiplied by itself four times, results in the original number. Since \(2^4 = 16\), we have \(\sqrt[4]{16} = 2\).

Step 2: Solve the Expression Inside the Parentheses

The original expression is \(\sqrt[4]{16} - 1 + 5^2\). First, simplify the expression \(\sqrt[4]{16} - 1\). We already found \(\sqrt[4]{16} = 2\), so we compute \(2 - 1 = 1\).

Step 3: Calculate the Power of 5

Next, calculate the power of 5, denoted by \(5^2\). This means multiplying 5 by itself: \(5 \times 5 = 25\).

Step 4: Final Calculation of the Expression

Now that we have \(1\) from the first part of the expression and \(25\) from \(5^2\), we add them together: \(1 + 25 = 26\).

Step 5: Verify the Result with a Calculator

Use a calculator to confirm that \(\sqrt[4]{16} - 1 + 5^2 = 2 - 1 + 25 = 26\). Check each part separately if necessary to ensure the accuracy of your result.

Key Concepts

Fourth Root

Understanding the concept of the fourth root is important because it forms the basic idea behind more complex mathematical operations. In our expression, \(\sqrt[4]{16}\) represents the fourth root of 16. Simply put, the fourth root of a number is a value which, when multiplied by itself four times, gives the original number. It's similar to how the square root squares the number only twice. Here, since \(2^4 = 16\), we find that \(\sqrt[4]{16} = 2\).

• Fourth root means multiplying the number by itself four times.
• In this exercise: \(2 \, \times \, 2 \, \times \, 2 \, \times \, 2 = 16\).
• Thus, \(\sqrt[4]{16} = 2\).

Learning the fourth root is not just about the number 16. The concept applies to all numbers. For instance, the fourth root of 81 is 3 because \(3^4 = 81\). Once you understand this, calculating any fourth root becomes much easier and faster.

Exponentiation

Exponentiation is a fundamental mathematical operation used extensively in algebra and calculus. In our exercise, we see this in \(5^2\), which simply means multiplying the number 5 by itself. As calculated, \(5 \times 5 = 25\). So, \(5^2 = 25\).

• Exponentiation is repeated multiplication of the same number.
• It is denoted by a small number (exponent) to the upper right of a base number.
• Here, "2" indicates that 5 should be multiplied by itself once more.

This operation allows simplification of expressions and can be easily extended to other bases and exponents. For instance, \(3^3 = 3 \, \times \, 3 \, \times \, 3 = 27\).Exponentiation adds a greater level of efficiency when dealing with large numbers, allowing easy computation of their powers.

Order of Operations

The order of operations is a critical concept when solving any arithmetic expressions, ensuring you get the right result. Known by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)), this rule dictates the sequence in which operations should be carried out.

• Start with calculations inside Parentheses.
• Proceed with Exponents.
• Then, tackle Multiplication and Division simultaneously, moving left to right.
• Finally, handle Addition and Subtraction, again from left to right.

In our exercise: \(\sqrt[4]{16} - 1 + 5^2\), the \(\sqrt[4]{16}\) is first simplified to 2; then \(5^2\) is computed to 25. According to PEMDAS, the subtraction is \(2 - 1 = 1 \) and lastly, adding the result to 25 gives 26.Understanding the correct order resolves potential mistakes in mathematics, as confusion between steps leads to incorrect responses.