Problem 41 - College Algebra with Modeling and Visualization

Question

Write the expression in radical notation. $$ 6^{1 / 2} $$

Step-by-Step Solution

Answer

\(6^{1/2}\) is \(\sqrt{6}\).

Step 1: Understand the Exponent

The expression given is \(6^{1/2}\). Here, the exponent \(\frac{1}{2}\) is the key part to translate the expression into radical notation. An exponent of \(\frac{1}{2}\) indicates a square root in radical notation.

Step 2: Translate to Radical Notation

To write the expression \(6^{1/2}\) in radical notation, you need to express it as a square root. In radical form, \(a^{1/n}\) is the \(n\)-th root of \(a\). Here, \(n = 2\), so the expression becomes \(\sqrt{6}\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation used to express repeated multiplication of a number by itself. For example, when you see something like \(a^b\), it means multiplying \(a\), the base, by itself \(b\) times. The exponent \(b\) tells you how many times to multiply the base.
One special case of exponentiation is when the exponent is a fraction. If the fraction is in the form \(\frac{1}{n}\), it means finding the \(n\)-th root of the base. This is because a fractional exponent represents a root.

• \(a^{1/2}\) represents the square root of \(a\).
• \(a^{1/3}\) represents the cube root of \(a\).

In the exercise, we had \(6^{1/2}\). Here, the exponent is \(\frac{1}{2}\), indicating a square root. Therefore, in radical notation, this becomes \(\sqrt{6}\). Understanding these nuances in exponentiation helps with converting expressions between exponential form and radical form.

Square Root

The square root is a specific type of root used frequently in mathematics. It is often encountered when dealing with powers and radicals.
The square root of a number \(a\), denoted as \(\sqrt{a}\), is a value that, when multiplied by itself, gives \(a\). For example, \(\sqrt{9} = 3\) because \(3 \times 3 = 9\). Square roots are important as they allow us to work with exponents and roots in different expressions.

• Remember that every positive number has two square roots: a positive and a negative one. However, the term "square root" typically refers to the positive root.
• The square root of a number is the same as raising that number to the power of \(1/2\).

In our specific exercise, the expression \(6^{1/2}\) is converted into \(\sqrt{6}\), emphasizing the concept of square roots appearing as fractional exponents.

Algebraic Expression

An algebraic expression is a combination of variables, numbers, and operations. It can include addition, subtraction, multiplication, division, and exponentiation, all of which make up a broad part of algebra.
Algebraic expressions can also include roots and radicals, which are often used in more advanced mathematics to express complex relationships. For instance, expressions like \(\sqrt{x}\) or \(x^{1/2}\) include both exponents and square roots.

• Expressions can be simplified or expanded based on algebraic rules. For example, converting \(6^{1/2}\) to \(\sqrt{6}\) is an example of simplifying using radical notation.
• Understanding how to manipulate these expressions is crucial for solving algebraic equations and understanding mathematical concepts.

The exercise illustrates how an algebraic expression in exponential form, \(6^{1/2}\), transitions into radical form, \(\sqrt{6}\), demonstrating fluency between different mathematical representations.