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Chapter 0: Problem 9

In Exercises \(1-28\), compute the numbers. $$ (16)^{1 / 2} $$

Short Answer

Expert verified
4

Step by step solution

01

- Understand the expression

The given expression is \( (16)^{1/2} \). This is asking for the square root of 16.
02

- Rewrite the expression

The exponent \( \frac{1}{2} \) can be rewritten as a square root. Thus, \( (16)^{1/2} \) is equivalent to \( \sqrt{16} \).
03

- Calculate the square root

Find the positive number that, when squared, equals 16. The square root of 16 is 4, because \( 4^2 = 16 \).

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Key Concepts

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

exponents
Exponents are a way to represent repeated multiplication of the same number. For instance, an exponent written as 3^2 means 3 multiplied by itself, giving us 9. In our exercise, we see an exponent written as 16^(1/2). This is a special exponent because it represents a fraction.
An exponent of 1/2 means that we are taking the number to the power of 1/2, which is equivalent to finding the square root of the number. More generally:
  • a^n where n is positive means multiply a by itself n times
  • a^(1/n) means take the n-th root of a
  • a^0 is always 1
  • a^(-n) means 1 divided by a^n
So, for the fraction exponent 16^(1/2), we need to find the value that squared (multiplied by itself) returns 16.
square root
A square root is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because 3 times 3 equals 9. It's written using the radical symbol, like this: \( \sqrt{9} \). In this exercise:
The square root is written as \( \sqrt{16} \), and we need to find which number multiplied by itself results in 16. We know:
\[ 4 \times 4 = 16 \] Hence, \( \sqrt{16} = 4 \).
Square roots have two kinds of results: positive and negative. The positive result is called the principal square root, which is what we typically look for in math exercises.
simplifying radical expressions
Simplifying radical expressions involves breaking down the numbers to their simplest form. In our exercise, we simplify \( (16)^{1/2} \) to \( \sqrt{16} \). Here鈥檚 how we do it step by step:
1. Recognize the exponent (1/2) as a square root.
2. Rewrite the expression to use the square root notation: \( (16)^{1/2} = \sqrt{16} \).
3. Calculate the simplified form by finding a number that, when squared, equals 16, which gives us 4.
Simplifying further radical numbers might involve factoring numbers into primes or utilizing perfect squares. It's a process which, ideally, makes the expression easier to work with or understand.

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