Problem 31 - Algebra 2 and Trigonometry

Question

In \(11-38,\) evaluate each expression in the set of real numbers. $$ \sqrt{x^{6}}, x \geq 0 $$

Step-by-Step Solution

Answer

The expression evaluates to \(x^3\) for \(x \geq 0\).

Step 1: Understand the problem

We need to find the value of the square root of the expression \(x^6\) where \(x\) is a non-negative real number (\(x \geq 0\)).

Step 2: Apply the property of square roots

Recall the property of square roots: \(\sqrt{a^2} = a\) for \(a \geq 0\). We can use this property to simplify \(\sqrt{x^6}\).

Step 3: Simplify the expression

We can rewrite \(x^6\) as \((x^3)^2\). Then, apply the square root property: \(\sqrt{(x^3)^2} = x^3\). This works because if \(x \geq 0\), then \(x^3 \geq 0\).

Step 4: Conclude the evaluation

Thus, the value of \(\sqrt{x^6}\) is simply \(x^3\), provided \(x \geq 0\).

Key Concepts

Understanding Square Roots

Square roots are foundational in algebra that allow us to find a number which, when multiplied by itself, gives us the original number. The square root is represented by the symbol \( \sqrt{} \). For example, the square root of 9 is 3 because \( 3 \times 3 = 9 \). The process basically "undoes" the squaring of a number.

When dealing with square roots, it is important that the number under the square root (called the radicand) is non-negative, especially when only considering real numbers. This restriction prevents any issues, such as getting imaginary numbers. For instance, \( \sqrt{16} = 4 \) is valid, while \( \sqrt{-16} \) would not fit within the set of real numbers.

• Key Property: For any non-negative real number \( a \), \( \sqrt{a^2} = a \).
• If \( a = x^3 \), then \( (a^2) = (x^3)^2 = x^6 \).
• Thus, \( \sqrt{x^6} = \sqrt{(x^3)^2} = x^3 \) when \( x \geq 0 \).

Exploring Exponents

Exponents describe how many times a number, known as the base, is multiplied by itself. The expression \( x^n \) refers to the base \( x \) raised to the power of \( n \). For example, \( x^6 = x \times x \times x \times x \times x \times x \). The power, \( n \), indicates how many instances we have of multiplying the base.

Exponents follow specific rules that help simplify expressions:

• Product Rule: \( a^m \times a^n = a^{m+n} \)
• Power of a Power Rule: \( (a^m)^n = a^{m \cdot n} \)
• Power of a Product Rule: \( (ab)^n = a^n \times b^n \)

These laws are key to rewriting and simplifying expressions with exponents effectively. For instance, the expression \( x^6 \) can be rewritten as \( (x^3)^2 \), which is useful when applying the properties of square roots. This makes handling more complex algebraic operations much simpler.

Understanding Real Numbers

Real numbers form one of the most crucial sets in mathematics, encompassing all the numbers that can be found on the number line. This set includes all the positive numbers, negative numbers, zero, and numbers with decimal points—making them crucial in algebra.

Real numbers can be categorized into subsets:

• Natural Numbers: Counting numbers starting from 1 (1, 2, 3, ...).
• Whole Numbers: Natural numbers including zero (0, 1, 2, 3, ...).
• Integers: Whole numbers as well as their negative counterparts (..., -3, -2, -1, 0, 1, 2, 3, ...).
• Rational Numbers: Numbers that can be expressed as a fraction of two integers (\( \frac{a}{b} \), where \( b eq 0 \)).
• Irrational Numbers: Numbers that cannot be written as simple fractions, such as \( \pi \) or \( \sqrt{2} \), often yielding non-repeating, non-terminating decimals.

When evaluating expressions like \( \sqrt{x^6} \) and determining it equals \( x^3 \) for \( x \geq 0 \), we rely on the rules governing real numbers to ensure the operation stays within bounds acceptable by this set.