Problem 58 - Algebra 2 and Trigonometry

Question

In \(58-73\) , write each power as a radical expression in simplest form. The variables are positive numbers. $$ 3^{\frac{1}{2}} $$

Step-by-Step Solution

Answer

The expression \( 3^{\frac{1}{2}} \) is equivalent to \( \sqrt{3} \).

Step 1: Understanding the Problem

We need to convert the given expression from an exponential form with a rational exponent to a radical form. The expression given is \( 3^{\frac{1}{2}} \).

Step 2: Recognizing the Exponent as a Root

When an expression is raised to a fractional power, such as \( a^{m/n} \), it can be written as a radical where the denominator \( n \) is the root. For the given expression \( 3^{\frac{1}{2}} \), the denominator is 2. This indicates we have a square root.

Step 3: Writing as a Radical Expression

The expression \( 3^{\frac{1}{2}} \) is equivalent to \( \sqrt{3} \). The denominator 2 tells us to take the square root. There are no further simplifications needed because 3 is a positive number already in simplest form.

Key Concepts

Understanding Exponentiation

Exponentiation is a mathematical operation involving two numbers: a base and an exponent.

• The base is the number that is being multiplied.
• The exponent represents the number of times the base is multiplied by itself.

For example, in the expression \(2^3\), 2 is the base and 3 is the exponent. It means that 2 is multiplied by itself 3 times: \(2 \times 2 \times 2 = 8\). Exponentiation is pivotal as it allows us to work efficiently with large numbers.
It is important to distinguish between integer and fractional exponents.

• Integer exponents are straightforward multiplications.
• Fractional exponents, often seen as rational numbers, involve roots which we will explore more in later sections.

Exploring Square Roots

The square root of a number is a value that, when multiplied by itself, gives back the original number.

• The square root is symbolized by the radical sign (\(\sqrt{}\)).
• For instance, \(\sqrt{9} = 3\) because \(3 \times 3 = 9\).

Square roots are especially relevant when dealing with exponential expressions having 1/2 as an exponent, since it transforms the expression into a square root.
In the context of our example, \(3^{\frac{1}{2}}\) translates to \(\sqrt{3}\).
This is because the denominator '2' in the exponent signifies and dictates the "root" operation we need to perform. Square roots simplify expressions while providing an alternative method of understanding and solving equations.
They help in simplifying radical expressions to their simplest form, which is crucial in algebraic problem-solving.

Mastering Rational Exponents

Rational exponents offer a bridge between exponents and radicals. They are defined as exponents that are fractions, where the numerator represents the power and the denominator represents the root of the number.

• An expression like \(a^{\frac{m}{n}}\) can be rewritten as \(\sqrt[n]{a^m}\) or \((\sqrt[n]{a})^m\).
• The numerator m indicates that the base is raised to that power.
• The denominator n directs us to the type of root taken.

In simpler terms, rational exponents bring together the concepts of raising a number to a power and taking a root in one clean step.
Consider our example \(3^{\frac{1}{2}}\), where 3 is the base:

• The numerator is 1, suggesting 3 stays as it is.
• The denominator is 2, indicating we take the square root.

This gives us \(\sqrt{3}\), showcasing how rational exponents facilitate the conversion between exponentiation and radical expressions elegantly. By mastering rational exponents, you can flexibly navigate between different forms of mathematical expressions and solve complex problems more effectively.