Problem 59 - Algebra 2 and Trigonometry

Question

In \(58-73\) , write each power as a radical expression in simplest form. The variables are positive numbers. $$ 5^{\frac{1}{2}} $$

Step-by-Step Solution

Answer

\(5^{\frac{1}{2}} = \sqrt{5}\)

Step 1: Recognize the Power

The expression given is \(5^{\frac{1}{2}}\), which is a fractional exponent.

Step 2: Convert the Power to a Radical

Recall that the fraction \(\frac{1}{2}\) in the exponent corresponds to a square root. Therefore, \(5^{\frac{1}{2}}\) can be rewritten as the square root of 5, which is \(\sqrt{5}\).

Step 3: Simplify

Since \(\sqrt{5}\) is already in its simplest form (as 5 is not a perfect square), no further simplification is needed.

Key Concepts

Understanding Fractional Exponents

Fractional exponents are an integral part of working with radical expressions. When you encounter an expression like \(a^{\frac{m}{n}}\), it's telling you to find the \(n\)-th root of \(a\) and then raise it to the \(m\)-th power. For example, \(a^{\frac{1}{2}}\) indicates the square root of \(a\). This method of expressing roots using exponents is useful in algebra for simplifying expressions and solving equations.

• \(a^{\frac{1}{2}}\) means the square root of \(a\)
• \(a^{\frac{1}{3}}\) means the cube root of \(a\)
• \(a^{\frac{2}{3}}\) means the cube root of \(a\) raised to the power of 2

Using fractional exponents allows for easier manipulation and simplification of expressions, especially when you need to multiply or divide powers. Understanding how to convert between fractional exponents and radical expressions is a valuable skill in simplifying mathematical problems.

Decoding the Square Root

The square root is one of the most commonly used radical expressions, represented by the symbol \(\sqrt{}\). Calculating a square root is the process of finding a number which, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because \(5 \times 5 = 25\). When you see a fractional exponent like \(\frac{1}{2}\), it implies taking the square root of a number. So, \(5^{\frac{1}{2}}\) becomes \(\sqrt{5}\). Remember the following points about square roots:

• Every positive number has a positive square root.
• Square roots of numbers that are not perfect squares (like 5, 7, 19) cannot be simplified into an integer.
• The square root of a perfect square like 9 or 16 results in integer values (i.e., \(\sqrt{9} = 3\)).

Knowing how to interpret and calculate square roots is crucial for simplifying expressions and solving quadratic equations, making it a vital concept in math.

Writing Expressions in Simplest Form

Simplifying expressions means to reduce them to their most basic or simplest form. This makes further calculations easier and more manageable. When dealing with radical expressions, reaching the simplest form often means converting complex formats into cleaner, more comprehensible versions. Take \(\sqrt{5}\) as our example here; since 5 is not a perfect square, \(\sqrt{5}\) is already in its simplest form. This is because there's no whole number whose square will give 5.To simplify expressions:

• Convert any fractional exponents to radical forms if possible.
• Check if the number under the square root is a perfect square.
• If it's not a perfect square, the expression is already simplified.

Simplifying to the simplest form is essential in math as it helps in focusing on the most important numerical relationships within an expression and avoids unnecessary complexity during problem-solving.