Problem 77 - Algebra 2

Question

Simplify each expression. \(x^{\sqrt{6}} \cdot x^{\sqrt{6}}\)

Step-by-Step Solution

Answer

The simplified expression is \(x^{2\sqrt{6}}\).

Step 1: Understand the Expression

The expression given is \(x^{\sqrt{6}} \cdot x^{\sqrt{6}}\). Both terms have the same base, \(x\), and exponents \(\sqrt{6}\). This is a multiplication of terms with the same base.

Step 2: Apply the Product of Powers Property

Use the product of powers property which states \(a^m \cdot a^n = a^{m+n}\). Here, \(a\) is \(x\), and both exponents \(m\) and \(n\) are \(\sqrt{6}\). Substitute these values to apply the property: \(x^{\sqrt{6}} \cdot x^{\sqrt{6}} = x^{\sqrt{6} + \sqrt{6}}\).

Step 3: Simplify the Exponent

Combine the exponents by adding them: \(\sqrt{6} + \sqrt{6} = 2\sqrt{6}\). Therefore, the expression simplifies to \(x^{2\sqrt{6}}\).

Key Concepts

Product of Powers

The product of powers rule is a valuable tool when dealing with expressions involving exponents. This rule simplifies the process of multiplying two powers that share the same base.
When you have a scenario like this, use the formula:

• \( a^m \cdot a^n = a^{m+n} \)

The base \( a \) remains the same, while the exponents \( m \) and \( n \) are simply added together. This allows you to combine the powers into a single expression, making it easier to work with.
Consider our expression: \( x^{\sqrt{6}} \cdot x^{\sqrt{6}} \). Here, the product of powers rule tells us to add the exponents, which leads to \( x^{2\sqrt{6}} \).
Remember this rule next time as it will save you time and effort in your algebraic calculations.

Simplification

Simplification is the process of reducing an expression to its most concise form. This is particularly helpful in mathematics because it makes equations more manageable and easier to understand.
In our case, we started with the expression \( x^{\sqrt{6}} \cdot x^{\sqrt{6}} \). To simplify, we applied the product of powers rule, which allowed us to combine the exponents: \( \sqrt{6} + \sqrt{6} = 2\sqrt{6} \).
Now, the expression becomes \( x^{2\sqrt{6}} \).

• This process highlights the simplification technique by transforming a complex expression into a singular, more straightforward form.

Simplification doesn't change the value of the expression, but it makes it easier to work with in subsequent mathematical operations.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and arithmetic operations. They are foundational in algebra and are used to represent relationships between quantities.
In our exercise, \( x^{\sqrt{6}} \cdot x^{\sqrt{6}} \) is an algebraic expression where:

• \( x \) is the variable, and
• \( \sqrt{6} \) is the exponent.

This expression represents the operation of exponentiation on the variable \( x \). Variables allow for flexibility as they can represent different numbers, and using them in expressions helps in generalizing mathematical ideas.
By simplifying this expression using algebraic rules like the product of powers, you refine the mathematical communication, losing nothing of the essential meaning, while gaining clarity and efficiency in problem-solving. Understanding algebraic expressions is crucial for advancing in algebra and other areas of mathematics.