Problem 45 - Algebra A Combined Function

Question

Find each root. Assume that all variables represent nonnegative real numbers. $$ \sqrt{81 x^{4}} $$

Step-by-Step Solution

Answer

The root of \( \sqrt{81 x^4} \) is \( 9x^2 \).

Step 1: Understand the Problem

The given problem asks us to find the roots of the expression \( \sqrt{81 x^{4}} \). Here, we need to simplify the square root to find an expression for the roots, assuming \( x \) is nonnegative.

Step 2: Simplify the Expression Inside the Square Root

The expression inside the square root is \( 81 x^4 \). We can break this down as the product of \( 81 \) and \( x^4 \). Thus, the expression becomes \( \sqrt{81} \cdot \sqrt{x^4} \).

Step 3: Find the Square Root of the Coefficient

Find the square root of the coefficient, which is 81. We express 81 as \( 9^2 \), so \( \sqrt{81} = 9 \).

Step 4: Simplify the Variable Exponent

Find the square root of \( x^4 \). Since \( x^4 = (x^2)^2 \), we have \( \sqrt{x^4} = x^2 \), assuming \( x \) is nonnegative.

Step 5: Combine the Results

Combine the results from Steps 3 and 4 to get the final root expression: \( 9x^2 \).

Key Concepts

Understanding Exponents

Exponents are a shorthand way of expressing repeated multiplication of a number by itself. In mathematical terms, we say that a number or variable, termed the base, is raised to an exponent or power. The exponent tells us how many times the base is used as a factor. For example, in the expression \(x^4\), the base \(x\) is multiplied by itself four times (\(x \times x \times x \times x\)).
This concept is crucial when working with square roots as they can often simplify the exponents in your expression.
When you take the square root of a number or expression, you are essentially looking for a value that, when multiplied by itself, equals the original number or expression. So, in the expression \(\sqrt{x^4}\), you are seeking a value that squared equals \(x^4\). The answer is \(x^2\), because \((x^2)^2 = x^4\).
Exponents help us in organizing, simplifying, and solving problems involving repeated multiplication. They are particularly handy when dealing with algebraic expressions involving powers of a variable.

Simplifying Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operators (like addition and subtraction). Examples of algebraic expressions include \(3x\), \(2x + 4\), or even \(\sqrt{81 x^4}\). Simplifying these expressions often involves breaking them down into simpler components, making them easier to work with.
When simplifying an algebraic expression like \(\sqrt{81 x^4}\), you can apply the property that the square root of a product is the product of the square roots. This allows you to split the expression into \(\sqrt{81}\) and \(\sqrt{x^4}\).
Simplifying further, you find that \(\sqrt{81} = 9\) (since \(81\) can be expressed as \(9^2\)), and \(\sqrt{x^4} = x^2\) (since \(x^4 = (x^2)^2\)).
The importance of simplifying expressions lies in its ability to transform complex expressions into simpler forms, making calculations more manageable and the relationships between elements clearer.

Nonnegative Real Numbers in Algebra

Nonnegative real numbers are the set of all positive numbers, including zero. This set is crucial in algebra, especially when dealing with square roots and exponential expressions. The square root function, by definition, only deals with nonnegative numbers because a square root of a negative number is not a real number; it results in what we call an imaginary number.
In exercises like finding \(\sqrt{81 x^4}\), it is often assumed that variables like \(x\) represent nonnegative real numbers. This assumption simplifies calculations by avoiding the need to consider negative values for \(x\). By operating within the realm of nonnegative numbers, you ensure that every square root calculation results in a real number.
Understanding the domain of nonnegative real numbers helps prevent errors in computation and ensures the validity of algebraic manipulations when solving equations.