Problem 38 - Algebra 2 and Trigonometry

Question

In \(38-57,\) write each radical expression as a power with positive exponents and express the answer in simplest form. The variables are positive numbers. $$ \sqrt{7} $$

Step-by-Step Solution

Answer

\( \sqrt{7} = 7^{\frac{1}{2}} \)

Step 1: Convert Radical to Exponent

The expression \( \sqrt{7} \) is a radical. The square root of a number can be expressed as the number raised to the \( \frac{1}{2} \) power. Thus, \( \sqrt{7} = 7^{\frac{1}{2}} \).

Step 2: Express in Simplest Form

Since \( 7^{\frac{1}{2}} \) is already expressed using positive exponents and cannot be simplified further, this is the simplest form of the expression.

Key Concepts

Positive Exponents

Exponents are a way to show how many times a number is multiplied by itself. When we talk about positive exponents, like in the expression \(7^{\frac{1}{2}}\), the exponent is positive, meaning that we are considering only the straightforward multiplication, not division or reciprocals. Positive exponents make expressions more manageable, especially when working with powers and roots. Whenever you deal with radicals (like square roots, cube roots), you can reframe them using positive exponents. This conversion simplifies many operations since working with exponents often reduces algebraic complexity.Here are a few guidelines:

• When you see a square root, think of it as a 1/2 power: \(\sqrt{a} = a^{\frac{1}{2}}\).
• A cube root corresponds to the 1/3 power: \(\sqrt[3]{a} = a^{\frac{1}{3}}\).
• For any \(n\)-th root, the exponent is \(\frac{1}{n}\), so \(\sqrt[n]{a} = a^{\frac{1}{n}}\).

Remember these conversions to switch between radical notation and exponential notation with positive exponents easily.

Simplest Form

The simplest form of an expression is when it cannot be simplified any further or reduced in complexity. Simplifying an expression is essential as it makes it more concise and easier to understand. In mathematical terms, it's like cleaning up a mess to see the essence clearly.When dealing with expressions like \(7^{\frac{1}{2}}\), determining the simplest form is about recognizing that no further simplification is possible. This expression is already in the simplest form because:

• The base 7 is a prime number and not suitable for further factorization.
• The exponent \(\frac{1}{2}\) can't be reduced or transformed into a simpler form without altering the meaning of the expression.

Therefore, expressions that are already using positive exponents and cannot be factored or simplified further are considered to be in their simplest form. It’s a clean and final way of representing a mathematical idea.

Converting Radicals to Exponents

Converting radicals to exponents is a powerful skill in algebra that helps in simplifying math expressions, especially when you need to apply rules of exponents. This conversion allows you to use exponent rules like multiplying powers more conveniently. To convert a radical to an exponent, follow these basic steps:

• Identify the root: Determine whether it is a square root, cube root, or any other \(n\)-th root.
• Express as a power: Write the expression with a fractional exponent corresponding to the root. For instance, a square root is equal to a power of \(\frac{1}{2}\).

For example, the square root \(\sqrt{7}\) converts to \(7^{\frac{1}{2}}\). This means the number 7 is raised to the power of \(\frac{1}{2}\), reflecting the square root operation.By converting radicals to exponents, you streamline calculations and can apply consistency across mathematical processes, especially in algebra, where such transformations are common.