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Chapter 7: Problem 16

Evaluate expression. \(16^{1 / 2}\)

Short Answer

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Step by step solution

01

Understand the Exponent

The given expression is written using a fractional exponent. The expression is: 16^{1 / 2} The fractional exponent 1/2 indicates a square root. So, we can rewrite the expression as: \[ 16^{1 / 2} \rightarrow \sqrt{16} \]
02

Calculate the Square Root

Determine the square root of 16. The square root of a number is a value that, when multiplied by itself, gives the original number. So, \[ \sqrt{16} = 4 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Root
A square root finds a number that, when multiplied by itself, equals the original number. For example, the square root of 16 is 4 because 4 multiplied by 4 equals 16: 4 \times 4 = 16. The symbol for the square root is \( \sqrt{} \). When you see an expression like \( \sqrt{16} \), it means 'what number squared equals 16?'. In our case, it's 4. Square roots are particularly useful for simplifying complex expressions and solving quadratic equations. It's crucial to grasp this concept as it frequently appears in various mathematical problems from algebra to calculus.
Evaluating Expressions
Evaluating expressions means finding the value of the expression. It often requires understanding the meaning of various symbols and following the correct order of operations.
For instance, in the given exercise, we evaluate \( 16^{1/2}\) by first recognizing that the exponent 1/2 signifies a square root.
The step-by-step process includes:
  • Identifying the type of operation: fractional exponent
  • Converting the fractional exponent to a root (e.g., 1/2 to square root)
  • Calculating the value of the root (e.g., square root of 16)
Using these steps ensures we correctly evaluate the expression and reach the right solution. This technique applies to various mathematical problems, making it essential for your math toolbox.
Exponentiation
Exponentiation is the process of raising a number to a power. It means multiplying a number by itself a certain number of times. For example, \(2^3\) equals 2 multiplied by itself three times: 2 \times 2 \times 2 = 8. In the given exercise, \(16^{1/2}\) involves a fractional exponent.
Fractional exponents provide an alternative way to express roots. \( 16^{1/2} \) is the same as \( \sqrt{16} \). What's crucial to understand is that \(a^{1/n}\) is the same as taking the nth root of a. For example:
  • \( 8^{1/3} \) = \( \sqrt[3]{8} \) = 2 (2 x 2 x 2 = 8)
  • \( 27^{1/3} \) = \( \sqrt[3]{27} \) = 3 (3 x 3 x 3 = 27)
Understanding exponentiation is key to mastering more complex algebraic and calculus concepts.

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