Solving and Understanding ( x^{1/2} cdot x^{-1/2} 3 ) and ( x^2 cdot x^{-1} 9 )

In the realm of algebra, understanding the manipulation of exponents is a fundamental skill. This article delves into the specifics of solving equations involving exponents, providing a step-by-step guide and reinforcing key concepts.

Introduction to Exponent Rules

Exponent rules are essential for simplifying and solving equations involving exponents. Some of the basic rules include the multiplication rule and the power of a power rule. This article will focus on how these rules can be applied to solve specific equations, such as those given in the problem statement.

Problem 1: ( x^{1/2} cdot x^{-1/2} 3 )

The first equation to solve is ( x^{1/2} cdot x^{-1/2} 3 ). Let's break this down step-by-step:

Square Both Sides: To simplify the equation, we can square both sides to eliminate the fractional exponents. Multiplying ( x^{1/2} cdot x^{-1/2} ) results in ( x^0 ), which equals 1. Therefore, squaring both sides gives: Result: ( (x^{1/2} cdot x^{-1/2})^2 3^2 ). Further Simplification: Simplifying the left side, we get ( (x^{1/2 - 1/2})^2 3^2 ), which reduces to ( 1^2 3^2 ). Thus, 1 9, which is incorrect. Let's revisit the initial steps: Step Correction: Revisiting the step where we simplify ( x^{1/2} cdot x^{-1/2} ), the correct value is actually 1, simplifying the equation to 1 9. Revisiting the Equation: Given that the base simplifies to 1, the equation becomes 1 9, which is a contradiction. Therefore, the provided equation ( x^{1/2} cdot x^{-1/2} 3 ) does not yield a valid solution. However, the correct interpretation and solution process for a similar equation are: Similar Case: If ( x^{1/2} cdot x^{-1/2} 1 ), then ( x^{1/2 - 1/2} 1 ), which is true for all ( x ). This demonstrates the importance of correctly applying exponent rules.

Problem 2: ( x^2 cdot x^{-1} 9 )

Next, let's solve the equation ( x^2 cdot x^{-1} 9 ) using algebraic manipulation:

Multiplication Rule: According to the multiplication rule of exponents, when multiplying two powers with the same base, we add their exponents: Apply the Rule: When we multiply ( x^2 cdot x^{-1} ), the equation simplifies to ( x^{2 - 1} 9 ). Result: This simplifies further to ( x^1 9 ), which is simply ( x 9 ). Verification: To verify, substitute ( x 9 ) back into the original equation: Substitution: ( 9^2 cdot 9^{-1} 9 cdot 1/9 1 cdot 9 9 ), confirming the solution is correct.

Doug Dillon’s Insight

Doug Dillon has provided a helpful solution to a similar problem, where if ( x^{1/n} cdot x^{-1/n} a ), then ( x^n cdot x^{-n} a ). This solution can be instructive for further understanding of exponent manipulation. Applying this to our equation, we can see that if ( x^2 cdot x^{-1} 9 ), then for ( x^2 cdot x^{-2} a ), the general form can be extended similarly:

General Form: For ( x^n cdot x^{-n} ), the result will always be 1, as ( x^{n - n} x^0 1 ). Applying the General Form: If we apply this general form to our specific case, we observe that this does not provide a direct solution but reinforces the understanding of exponent rules.

Conclusion

In conclusion, understanding and correctly applying the exponent rules is crucial for solving algebraic equations involving exponents. By breaking down the problems step-by-step and carefully applying the rules, we can simplify and solve these equations. The specific examples provided demonstrate the importance of carefully handling each step to achieve a coherent and accurate solution.

Keywords: exponent rules, algebraic equations, solving equations