Problem 58 - Intermediate Algebra

Question

Simplify each radical. Assume that all variables represent positive real numbers. $$ \sqrt{16 x^{8}} $$

Step-by-Step Solution

Answer

The simplified radical is \( 4x^4 \).

Step 1: Identify the Components

First, let's break down the expression to understand each part. The expression is \( \sqrt{16x^8} \), which contains a constant term 16 and a variable term \( x^8 \). We know that both are under the square root.

Step 2: Simplify the Constant Term

The square root of 16 is 4, because \( 4^2 = 16 \). Thus, we can simplify the constant term to 4 outside the square root.

Step 3: Simplify the Variable Term

For \( x^8 \), we use the property \( \sqrt{x^{2n}} = x^n \). Here, \( x^8 = (x^4)^2 \), so \( \sqrt{x^8} = x^4 \).

Step 4: Combine Results

Now that we have simplified both components, combine them. The simplified expression is \( 4x^4 \).

Key Concepts

Square Roots

Square roots are a fundamental concept in mathematics, particularly in relation to radicals. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, since multiplying 4 by 4 equals 16.

• Notation: The square root of a number is denoted by the radical symbol \(\sqrt{}\). For instance, \(\sqrt{16}\) represents the principal square root of 16.
• Principal Square Root: Generally, when we refer to the square root of a positive number, we are referring to its principal square root, which is always positive. In our example, \(\sqrt{16}\) simplifies to 4, not -4, even though \((-4) \times (-4) = 16\).

Understanding square roots helps in simplifying more complex expressions, breaking them down into manageable parts. It’s crucial to identify parts of a square root separately, especially when dealing with expressions involving variables, as in \(\sqrt{16x^8}\). In such expressions, separate the numerical and variable parts to simplify using square root rules.

Exponent Rules

Exponent rules are essential for simplifying expressions that involve powers of variables. They tell us how to handle operations on numbers with exponents.

• Product of Powers: When multiplying like bases, add the exponents. For example, \(x^a \times x^b = x^{a+b}\).
• Power of a Power: When raising an exponent to another power, multiply the exponents: \((x^a)^b = x^{ab}\).
• Power of a Product: Distribute the exponent to each factor inside the parentheses: \((xy)^a = x^a y^a\).
• Simplifying Radicals: When simplifying expressions under a square root, using the rule \(\sqrt{x^{2n}} = x^n\) can be helpful. It allows us to take paired factors from under the square root.

In the original exercise, understanding these rules is crucial. For instance, for \(x^8\) under the square root, recognizing \(x^8 = (x^4)^2\) allows us to simplify \(\sqrt{x^8}\) to \(x^4\). This usage of the power of a power rule helps translate a potentially complicated expression into something simpler.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, etc. Simplifying a radical expression requires a solid grasp of both square roots and exponent rules. With these, you can deconstruct and simplify radicals confidently.

• Breaking Down Radicals: When faced with a composite number or a variable under a radical, break it down into known factors which can be managed separately.
• Combining Like Terms: When simplifying, always combine similar terms, simplifying both constants and variables separately using rules of exponents.
• Radicals in Algebra: In algebraic expressions containing radicals, identify opportunities to simplify each part using the properties of roots and powers.

In the case of \(\sqrt{16x^8}\), the process involved simplifying 16 to 4 using \(\sqrt{16} = 4\), and \(x^8\) to \(x^4\) using the property \(\sqrt{x^{8}} = x^{4}\). Breaking down each component before recombining them is the key to handling radical expressions effectively. These steps enhance clarity and simplify complex expressions, which is essential for problem-solving in mathematics.