Problem 11 - Elementary and Intermediate Algebra

Question

Assume for all exercises that even roots are of non- negative quantities and that all denominators are nonzero. Write an equivalent expression using radical notation and, if possible, simplify. $$ 16^{1 / 2} $$

Step-by-Step Solution

Answer

4

Step 1: Understand the Expression

The given expression is \(16^{1/2}\). This expression involves an exponent, which represents a power or root in this case.

Step 2: Convert the Rational Exponent to Radical Notation

An exponent of \(1/2\) indicates a square root. Therefore, \(16^{1/2}\) can be rewritten using radical notation as \(\sqrt{16}\).

Step 3: Simplify the Radical Expression

The square root of 16 is 4 because \(4^2 = 16\). Thus, \(\sqrt{16} = 4\).

Key Concepts

Rational Exponents

Rational exponents can seem confusing at first, but they're really just another way to write roots. When you see an exponent that's a fraction, like in the expression \( 16^{1/2} \), it means you're dealing with a root. The denominator of the fraction (in this case, 2) signifies the type of root. Specifically, \( 16^{1/2} \) translates to the square root of 16.

To generalize:

• For any number \( a \) and any denominator \( n \) in the exponent \( a^{1/n} \), it means you're finding the \( n^{th} \) root of \( a \).

Square Root

A square root is a special type of radical expression. It asks the question: what number squared (multiplied by itself) equals the given number? For our example, \( 16^{1/2} \), we're looking for the square root of 16.

The square root of 16 is written as \( \sqrt{16} \). To find the answer, consider which number multiplied by itself gives 16. The number is 4, because \( 4 \times 4 = 16 \). Therefore, \( \sqrt{16} = 4 \).

Simplifying Expressions

Simplifying expressions means breaking them down into their simplest form. Let's go through how we do this with our example, \( 16^{1/2} \).

• First, identify that the exponent \( 1/2 \) indicates a square root.


• Next, rewrite the expression using radical notation, turning \( 16^{1/2} \) into \( \sqrt{16} \).


• Finally, simplify \( \sqrt{16} \). Since \( 4 \times 4 = 16 \), \(\sqrt{16} \) simplifies to 4.

So, the simplified form of \( 16^{1/2} \) is 4.

It's important to practice these steps to become comfortable with switching between rational exponents and radical notation, as well as simplifying the resulting expressions.