Problem 25 - Elementary Algebra

Question

Simplify each expression by removing the radical sign. Assume each variable is nonnegative. $$ \sqrt{x^{2 n}} \text { , where } n \text { is a natural number. } $$

Step-by-Step Solution

Answer

Question: Simplify the expression: $\sqrt{x^{2 n}}$ Answer: The simplified expression is: $x^n$

Step 1: Express the given expression as a power

First, we will rewrite the square root as a power to its simplest form: $$ \sqrt{x^{2 n}} = x^{\frac{2n}{2}} $$

Step 2: Simplify the exponent

Now, we will simplify the exponent by dividing 2n by 2: $$ x^{\frac{2n}{2}} = x^n $$

Step 3: Final Simplified Expression

The final simplified expression is: $$ \sqrt{x^{2 n}} = x^n $$ For any nonnegative variable x and natural number n.

Key Concepts

Radical Expressions

A radical expression is an expression that includes a root symbol, such as a square root or cube root. In the case of a square root, the notation \(\sqrt{x}\) is used, where \(x\) represents the radicand - the number or expression under the root symbol. Square roots are a common type of radical expression, especially when dealing with algebraic expressions. They allow us to find a number which, when multiplied by itself, gives the original radicand. For example, the square root of 9 is 3, because \(3 \times 3 = 9\).

When simplifying radical expressions, it is often helpful to express the root as an exponent because this can facilitate simplification. For example, \(\sqrt{x}\) can be rewritten as \(x^{1/2}\). In this form, it becomes easier to manipulate the expression and combine it with other algebraic terms. Understanding how to work with radical expressions is crucial as they appear frequently in algebra and higher-level mathematics, helping in solving equations and simplifying expressions.

Exponents

Exponents are a foundational concept in algebra that represent repeated multiplication. When a number or variable is raised to an exponent, it means that number is multiplied by itself a specific number of times. For example, \(x^3\) means \(x \times x \times x\).

Working with exponents involves certain rules that simplify computation, including the power of a power, product of powers, and quotient of powers rules.

• The product of powers rule states that \(a^m \times a^n = a^{m+n}\).
• The quotient of powers rule says \(a^m / a^n = a^{m-n}\) as long as \(a eq 0\).
• The power of a power rule is \((a^m)^n = a^{m \times n}\).

These rules are essential when simplifying expressions with exponents, such as in the step from \(\sqrt{x^{2n}}\) to \(x^n\) by rewriting the square root as \(x^{2n/2}\), and then simplifying it to \(x^n\).

Mastering exponents is key to solving more complex algebraic expressions and understanding exponential growth, a concept that appears frequently in mathematics as well as real-world applications.

Simplification

Simplification is the process of reducing an algebraic expression to its simplest form, making it easier to read and work with. This involves using mathematical rules and operations to remove redundant components, combine like terms, and transform expressions into a form that's easier to handle.

For example, consider the expression \(\sqrt{x^{2n}}\). Here, the goal is to remove the radical sign and simplify the expression. This process involves first rewriting the square root as a fractional exponent \(x^{2n/2}\) which further simplifies to \(x^n\) by dividing the exponent \(2n\) by 2.

Simplifying expressions is a skill that improves problem-solving efficiency and understanding. It involves deciding on the right operations to apply at each step and ensures that calculations are as straightforward as possible. This is not only useful within algebra but is a vital skill across all areas of mathematics and its applications.