Problem 80 - Algebra 2

Question

Simplify each radical expression. Use absolute value symbols as needed. $$ \sqrt{0.25 x^{6}} $$

Step-by-Step Solution

Answer

The simplified radical expression is \(0.5|x^{3}|\).

Step 1: Simplify the numeric part of the expression

First, let's simplify the numeric part inside the square root. Since \(\sqrt{0.25}\) is \(0.5\), the expression now looks like \(0.5\sqrt{x^{6}}\).

Step 2: Simplify the variable part of the expression

Next, let's tackle the variable part, \(\sqrt{x^{6}}\). Recall that the square root of \(x^{2}\) is |x|. We can write \(x^{6}\) as \((x^{3})^{2}\), which simplifies into \(|x^{3}|\).

Step 3: Combine the results

Finally, combining the results from Step 1 and Step 2 gives the answer \(0.5|x^{3}|\).

Key Concepts

Understanding Absolute Value

In mathematics, the absolute value of a number is its distance from zero on the number line, regardless of direction. This means that the absolute value is always non-negative. When dealing with expressions that involve variables, the absolute value ensures that the result is positive, regardless of the value of the variable.
For example:

• The absolute value of 5 is 5, because it's 5 units away from zero.
• The absolute value of -5 is also 5, because we ignore the negative sign.

In expressions, especially those involving radicals or exponents, you might see the absolute value to indicate the principal square root. For example, when simplifying \(\sqrt{x^2}\), the result is \(|x|\), because it reflects a non-negative output.
This is crucial in maintaining the correctness of mathematical expressions, especially when working with odd or even powers.

The Nature of Square Roots

Square root calculations are fundamental in algebra. A square root of a number \(a\) is a value that, when multiplied by itself, gives \(a\). For instance, the square root of 9 is 3, as \(3 \times 3 = 9\). Square roots are typically denoted by the radical symbol \(\sqrt{}\).
Understanding how to simplify square roots, especially those involving variables, is a critical skill:

• When simplifying \(\sqrt{a^2}\), the result is \(|a|\), because it represents the positive root.
• For more complex expressions, like \(\sqrt{x^6}\), which can be rewritten using exponent rules as \((x^3)^2\), the square root becomes \(|x^3|\).

This simplification helps in ensuring that expressions remain manageable and accurate, especially when part of larger mathematical operations.

Working with Exponents

Exponents express repeated multiplication of a base number. For example, \(x^3\) means \(x \times x \times x\). They're a versatile tool in algebra, allowing for the simplification of expressions and solving equations efficiently.
Some useful points about exponents include:

• Raising a power to another power involves multiplying the exponents, such as \((x^3)^2 = x^6\).
• When taking the square root of a power, such as \(\sqrt{x^6}\), you can think of it as raising \(x^6\) to the power of \(\frac{1}{2}\), resulting in \(x^3\).

Understanding these properties allows you to manage and transform expressions with ease, enabling solutions to more complex problems.

Exploring Variable Expressions

Variable expressions contain numbers, operations, and variables. Variables are symbols, usually letters, that stand in for unknown values or quantities. Mastering variable expressions involves understanding how operations like addition, subtraction, multiplication, division, and exponents affect them.
With expressions like \(\sqrt{0.25x^6}\), each component, including the variable part, needs careful handling:

• First, decipher the fixed numeric portion, such as \(\sqrt{0.25} = 0.5\).
• Next, address the variable component using knowledge of exponents and roots, leading to simplified results like \(|x^3|\).

This steady step-by-step simplification helps in ending up with a neat result that embodies the original expression's intent, ensuring clarity and correctness.