Problem 21 - Algebra 2

Question

Write each radical using rational exponents. $$ \sqrt{23} $$

Step-by-Step Solution

Answer

\( \sqrt{23} = 23^{1/2} \).

Step 1: Identify the Radical Form

The given expression is \( \sqrt{23} \), which is a square root.

Step 2: Understand the Conversion to Rational Exponents

We need to express the square root in terms of rational exponents. The square root of a number \( x \) is equivalent to \( x^{1/2} \).

Step 3: Rewrite Using a Rational Exponent

Apply the conversion to the problem: \( \sqrt{23} = 23^{1/2} \).

Key Concepts

Radical Expressions

Radical expressions are mathematical expressions that involve roots, such as the square root, cube root, and so on. These expressions often include a radical sign, which looks like a check mark (√). The number under the radical sign is called the radicand.
For example, in the expression \(\sqrt{23}\), 23 is the radicand. Radical expressions are a broad category that includes various types of roots:

• Square roots
• Cube roots
• Fourth roots, and more.

These expressions are useful in many mathematics fields, including algebra and calculus. They help simplify equations and can give approximate or exact solutions to problems.

Square Roots

A square root is a specific type of radical expression where the root is 2, which is the implied number if no index number is shown. Essentially, finding a square root means identifying what number, when multiplied by itself, equals the given radicand.
For instance, the square root of 25 is 5, because 5 times 5 equals 25. Similarly, the square root of 23, expressed as \(\sqrt{23}\), is the number that when multiplied by itself gives approximately 23. However, 23 is not a perfect square, meaning it doesn't have an integer square root.
Square roots are particularly useful in solving quadratic equations and appear frequently in geometry, especially in dealing with right-angled triangles through the Pythagorean theorem.

Converting Radicals to Exponents

Converting radicals to exponents involves expressing a radical expression in terms of rational exponents. This conversion is beneficial because it allows us to use the properties of exponents, such as the rules for multiplication and division, which can simplify complex expressions.
In the case of square roots, converting involves expressing the square root of a number \(x\) as \(x^{1/2}\). For example, \(\sqrt{23}\) can be written as \(23^{1/2}\). This equivalence is particularly handy when performing operations with radicals, such as multiplication or division, or when integrating into more complex algebraic equations.

• It simplifies the operation by turning roots into exponents.
• Helps in easier computation and manipulation of the expression.

Understanding this conversion provides a powerful tool for tackling mathematical problems across different domains.