Problem 15 - Algebra A Combined Function

Question

Find each square root. Assume that all variables represent nonnegative real numbers. $$ \sqrt{x^{10}} $$

Step-by-Step Solution

Answer

The square root of \(x^{10}\) is \(x^5\).

Step 1: Understand the concept of square root

The square root of a number or expression is a value that, when multiplied by itself, gives the original number or expression. In our case, it’s important to remember that when taking the square root of an expression with an exponent, the rule is to divide the exponent by 2.

Step 2: Apply the square root rule to the exponent

Given the expression \(\sqrt{x^{10}}\), we will apply the rule for the square root of exponents. This means we will divide the exponent 10 by 2, because the square root of \(x^n\) is \(x^{n/2}\).

Step 3: Calculate the new exponent

Divide the exponent 10 by 2 to get \(10/2 = 5\). This results in a new expression: \(x^5\).

Step 4: Write the final expression

After calculating, the square root of \(x^{10}\) simplifies to \(x^5\). Thus, the expression \(\sqrt{x^{10}}\) equals \(x^5\).

Key Concepts

Exponents

Exponents are a mathematical notation used to express how many times a number, known as the base, is multiplied by itself. In the expression \(x^{10}\), the number 10 is the exponent, and it tells you to multiply \(x\) by itself 10 times. Exponential notation is a more efficient way of writing lengthy multiplications.

• For example, \(x^3\) means \(x \times x \times x\).
• The rules of exponents assist in simplifying calculations, especially in algebraic expressions.

When finding the square root of an expression with an exponent, it's essential to remember a key rule: the square root of \(x^n\) is \(x^{n/2}\). This rule simplifies the process of finding square roots in algebraic expressions and can make complex problems more manageable.

Real Numbers

Real numbers include all the numbers on the number line. This set is made up of both rational and irrational numbers, integrating all positive and negative integers, fractions, and decimals.

• Rational Numbers: These can be expressed as a fraction or decimal, like 1/2 or 0.5.
• Irrational Numbers: Numbers that cannot be expressed as a simple fraction, such as \(\pi\) or \(\sqrt{2}\).

In the context of the exercise, the variable \(x\) is assumed to be a non-negative real number. This is important when working with square roots, as the square root function is defined for non-negative input from real numbers. This ensures that the outcome is also a real number.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (like addition, subtraction, multiplication, and division). They are the building blocks of algebra and are used to solve various mathematical problems.For example, an expression like \(3x + 2\) shows how numbers (2 and the implied 3) and variables (\(x\)) interact. When dealing with algebraic expressions involving square roots, it's essential to simplify them properly, often by applying the rules of exponents.

• In our example, \(\sqrt{x^{10}}\) simplifies to an algebraic expression \(x^5\), as calculated using exponent rules.
• Understanding how to manipulate and simplify these expressions is fundamental in algebra, making it easier to solve equations and perform other mathematical operations.

This clarity in dealing with algebraic expressions aids in breaking down complex problems to manageable parts, making it easier to find solutions.