Problem 55 - Precalculus Mathematics for Calculus

Question

Simplify the expression. Assume that the letters denote any real numbers. $$\sqrt[4]{16 x^{8}}$$

Step-by-Step Solution

Answer

The simplified expression is \( 2x^2 \).

Step 1: Recognize the Expression

The expression given is \( \sqrt[4]{16 x^{8}} \), which means we need to find the fourth root of the entire expression inside the radical.

Step 2: Simplify Inside the Radical

The expression inside the radical is \( 16 x^{8} \). Notice that 16 can be rewritten as \( 2^{4} \) and \( x^{8} \) is already a power that can be simplified when taking the fourth root.

Step 3: Apply the Fourth Root

We can now apply the fourth root to both components separately: \( \sqrt[4]{2^4} \) is the fourth root of \( 2^4 \), which is \( 2 \), since it's like taking the power 1 or \( 2^{4/4} = 2^1 \). For \( \sqrt[4]{x^8} \), applying the fourth root gives \( (x^8)^{1/4} = x^2 \), because \( 8/4 = 2 \).

Step 4: Combine the Results

Now combine both results: The fourth root of \( 16x^8 \) is \( 2 \times x^2 = 2x^2 \).

Key Concepts

Fourth Roots

Taking the fourth root of an expression means finding a number that, when multiplied by itself four times, gives the original expression. In simpler terms, if you have a fourth root, such as \( \sqrt[4]{a} \), this represents a number that satisfies \( x^4 = a \). This is particularly straightforward if you deal with perfect powers, like \( 16 \) which is \( 2^4 \).

• The fourth root of \( 16 \) is \( 2 \) because multiplying \( 2 \times 2 \times 2 \times 2 = 16 \).
• To find fourth roots of more complex expressions, apply the root separately to each term.

Breaking down expressions to their basic terms aids in understanding how roots apply across different terms, especially when simplifying radical expressions.

Exponents

Exponents are a shorthand way to express repeated multiplication. For example, \( x^8 \) implies that \( x \) is multiplied by itself eight times. When simplifying expressions involving exponents, you often need to use the rules of exponents, such as the quotient rule or power rule.

• The power rule states that \( (x^m)^n = x^{m \times n} \).
• This is crucial when dealing with radicals and roots, as converting roots to powers makes computation simpler.

In the solution provided, \( x^8 \) was simplified using the rule: \( (x^8)^{1/4} = x^{8 \times (1/4)} = x^2 \). Applying these principles can significantly reduce complexity in algebraic expressions.

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and fourth roots. They require careful simplification to make them as simple as possible. Simplifying a radical expression often involves finding perfect powers within the radical.

• The goal here is to simplify the expression inside the radical so that taking the root becomes straightforward.
• For instance, changing \( 16 \) to \( 2^4 \) makes it easier to apply the root.

Understanding how to manipulate expressions inside radicals, such as turning them into perfect powers, helps in efficiently simplifying complicated radical expressions.

Real Numbers

Real numbers include all the numbers you can think of on the number line. This includes positive numbers, negative numbers, and zero, covering both rational numbers like \( \frac{1}{2} \) and irrational numbers like \( \sqrt{2} \).

• Real numbers are foundational in algebra because they give context to the values you work with in equations.
• In simplifying expressions, particularly in determining roots, knowing that all letters denote real numbers ensures valid operations throughout the solution process.

Whenever an expression includes symbols representing any real numbers, it implies the expression retains valid for each possible real value, allowing for broad and flexible application.