Problem 39 - Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

The root is \( 9x^2 \).

Step 1: Identify the Problem Components

We need to simplify the expression \( \sqrt{81x^4} \). The components to examine include the numerical part \( 81 \) and the variable part \( x^4 \).

Step 2: Simplify the Numerical Part

Find the square root of the number \( 81 \). We know that \( 81 = 9^2 \), so the square root of \( 81 \) is \( 9 \).

Step 3: Simplify the Variable Part

Consider the variable part \( x^4 \). The square root of \( x^4 \) is \( x^2 \) because \( (x^2)^2 = x^4 \).

Step 4: Combine the Simplified Parts

Combine the simplified numerical and variable parts. The square root of \( 81x^4 \) is \( 9x^2 \).

Key Concepts

Simplifying Radicals

When simplifying radicals, the goal is to reduce the expression inside the radical sign to its simplest form. Radical expressions often involve roots, with square roots being the most common. To simplify the square root of a number or monomial, follow these steps:

• Find the prime factorization of the number inside the radical.
• Pair the factors whenever possible, since the square root of a pair of equal factors is just one factor.
• Repeat the process for each part, including both numbers and variables, of the radical expression.

For example, with the expression \(\sqrt{81x^4}\), discretely simplify both the numeric and the variable components.`9` is the square root of `81`, and `x^2` is the square root for `x^4`; hence the radical simplifies to \(9x^2\).
Understanding how to break down these components helps clarify the structure of more complex expressions, allowing you to simplify them efficiently.

Exponents

Exponents are a way to express repeated multiplication, showing how many times a number, known as the base, is multiplied by itself. Exponents are crucial in working with radicals, as taking roots is essentially finding which number raised to a certain power gives the original number.
The properties of exponents greatly assist in simplifying expressions:

• A power of a power means you multiply the exponents (e.g., \(a^{m })^{n} = a^{m \times n}\)).
• Multiplying like bases means you add the exponents (e.g., \(a^m \times a^n = a^{m+n}\)).
• Division of like bases means you subtract the exponents (e.g., \(a^m \div a^n = a^{m-n}\)).

In the case of \(\sqrt{81x^{4}}\), the simplification \(x^{4} = (x^{2})^{2}\) uses the power of a power property to reduce the exponent when taking the square root. This understanding makes breaking down more intricate algebraic expressions straightforward.

Algebraic Expressions

An algebraic expression is composed of variables, numbers, and operations (like addition, subtraction, multiplication, and division). Simplifying these expressions is central to solving algebra problems effectively. By breaking them into recognizable parts, such as combining like terms or using the distributive property, they become more manageable.
In algebraic expressions with radicals, simplification is key to achieving cleaner results:

• Identify similar terms that can be combined.
• Use the rules of operations (such as distribution or combination) to rewrite the expression.
• Simplify expressions involving operations and radicals by employing properties of exponents and roots.

When tackling \(\sqrt{81x^4}\), separate numbers and variables, then simplify each part independently before recombining them. The result \(9x^2\) demonstrates how simplifying radicals within algebraic expressions maintains the integrity of the expression while making it easier to interpret and handle.