Problem 9 - Intermediate Algebra

Question

Simplify. Assume that variables represent positive real numbers. $$ \sqrt{x^{10}} $$

Step-by-Step Solution

Answer

The simplified form is \( x^5 \).

Step 1: Identify the expression under square root

The given expression is \( \sqrt{x^{10}} \). The goal is to simplify this expression by using the laws of exponents and roots.

Step 2: Use the property of square roots

Recall the property of square roots: \( \sqrt{a^b} = a^{b/2} \). We apply this property to the expression \( x^{10} \).

Step 3: Simplify the exponent

Applying the square root property gives us: \( \sqrt{x^{10}} = x^{10/2} = x^5 \). Thus, the expression simplifies to \( x^5 \).

Key Concepts

Square Roots

Square roots are mathematical operations that identify a value which, when multiplied by itself, results in the original number. For example, the square root of 9 is 3, because 3 multiplied by itself is 9. In mathematical terms, we write the square root of a number \( a \) as \( \sqrt{a} \). Square root operations simplify expressions but require a good grasp of square root properties:

• The square root of a product is the product of the square roots, \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \).
• The square root of a quotient is the quotient of the square roots, \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \).
• When a number is squared (multiplied by itself) and then square-rooted, you return to the original number, if it’s positive, \( \sqrt{a^2} = a \) assuming \( a \) is a positive real number.

These properties allow us to understand expressions like \( \sqrt{x^{10}} \) better. We use the square root property \( \sqrt{a^b} = a^{b/2} \) to simplify such expressions. This helps greatly in reducing complex expressions to simpler forms.

Laws of Exponents

The laws of exponents are essential for handling expressions with powers effectively and accurately. When you see an expression like \( x^{10} \), it's important to remember these fundamental rules:

• Product Rule: \( a^m \cdot a^n = a^{m+n} \).
• Quotient Rule: \( \frac{a^m}{a^n} = a^{m-n} \) where \( a eq 0 \).
• Power Rule: \( (a^m)^n = a^{m \cdot n} \).
• Root Rule: This is particularly critical for our example \( \sqrt{x^{10}} \), because \( \sqrt{a^b} = a^{b/2} \).

Using these helps simplify complex expressions like square roots or powers by reducing them to an easier to manage format. Simplifying \( \sqrt{x^{10}} = x^{10/2} = x^5 \) uses the root rule, cutting through the complexity by dividing the original power by two.

Positive Real Numbers

In mathematical expressions, the assumption that variables represent positive real numbers is crucial. Positive real numbers are any number greater than zero, and they include fractions, integers, and irrational numbers but not negative numbers or zero. This assumption simplifies working with expressions, particularly those involving roots and exponents:

• Square Roots: For square roots, this ensures that you can always find a real number result. For instance, \( \sqrt{x^{10}} \) is simplified to \( x^5 \) safely, as \( x \) is positive.
• Exponents: Raising a positive real number to any power remains positive, so you never have to deal with unexpected negative results in root operations.
This concept helps ensure calculations remain within the domain of real numbers, avoiding complex number situations unless specifically required. In exercises like simplifying \( \sqrt{x^{10}} \), assuming \( x \) is a positive real number means we can perform operations like division of exponents without concerns over negative or complex outcomes.