Problem 42 - Algebra 2 and Trigonometry

Question

In \(39-42,\) find the set of real numbers for which the given radical is a real number. $$ \sqrt[4]{x+5} $$

Step-by-Step Solution

Answer

The set of real numbers is \([-5, \infty)\).

Step 1: Understand the Radical Expression

The expression given is \( \sqrt[4]{x+5} \). This represents the fourth root of \( x+5 \). For a real number result, \( x+5 \) needs to be non-negative, because the fourth root of a negative number would not be real.

Step 2: Set Up the Inequality

To ensure that \( \sqrt[4]{x+5} \) is real, we must have \( x+5 \geq 0 \). This inequality ensures that \( x+5 \) is non-negative.

Step 3: Solve the Inequality

Solve the inequality \( x+5 \geq 0 \) by isolating \( x \). Subtract 5 from both sides to get \( x \geq -5 \).

Step 4: Write the Solution Set

The solution to the inequality \( x \geq -5 \) means that any real number greater than or equal to \(-5\) is valid. Therefore, the set of real numbers for which the radical is real is \([ -5, \infty )\).

Key Concepts

Inequality solutions

When working with radical expressions in algebra, inequalities are essential for understanding which values make the expression valid. For example, consider an expression like \( \sqrt[4]{x+5} \). To find values where the expression results in a real number, we need to ensure \( x+5 \) is non-negative.

Why is this important? Because real solutions exist only when the value within the radical, which in this case is \( x+5 \), is zero or positive, as radicals of negative values result in non-real numbers in this context. This leads to setting up an inequality:

• \( x+5 \geq 0 \)

By solving this inequality, we isolate \( x \) to identify the range of valid real numbers. Remove 5 from both sides to derive \( x \geq -5 \). This tells us that for any \( x \) equal to or greater than -5, the radical expression will hold a real number solution.

Real numbers

Real numbers are a fundamental concept in algebra, represented by all numbers on the continuous number line including integers, fractions, and irrational numbers. When dealing with radicals, especially with terms under the fourth root, understanding where real numbers lie is critical.

In the context of an expression like \( \sqrt[4]{x+5} \), ensuring you have a real number means checking that the expression inside the radical (\( x+5 \)) is not negative. Why does this matter? Because the fourth root of negative numbers cannot be represented by real numbers — they would instead result in complex or imaginary numbers.

Thus, to ensure that the entire expression evaluates to a real number, initially setting the inequality \( x+5 \geq 0 \) and solving it highlights where the valid real numbers begin, i.e., \( x \geq -5 \). Therefore, the solution set contains all numbers starting from \(-5\) and extending positively towards infinity, expressing our results as the interval \([-5, \, \infty)\).

Fourth root

The concept of the fourth root in mathematics involves identifying a number which, when taken to the power of four, gives the original number under the root. For example, in \( \sqrt[4]{x+5} \), we are seeking a number that, when raised to the fourth power, equals \( x+5 \).

Understanding the mechanics of the fourth root is crucial because it influences which values \( x \) can take.

• A fourth root is similar in nature to a square root but instead involves a factor of four.
• To ensure a valid fourth root in real numbers, the expression within the radical must be non-negative.

In practice, solving for conditions where \( \sqrt[4]{x+5} \) remains real prompts us to establish that \( x+5 \geq 0 \) and consequently solve for \( x \). Here, the solutions form a clear path by deriving \( x \geq -5 \), effectively encompassing the condition needed for realistic fourth root computation. By adhering to these guidelines, you ensure a consistent and accurate application of radicals in algebra, securing clear and real results.