Simplify and Solve Mathematical Expressions Involving Exponents
When working with mathematical expressions, understanding how to simplify and solve them is a crucial skill. In this article, we will solve the given expression: 5^{-3/2} 5^{-1/2} 5^{1/2}. We will break down the process step-by-step to ensure you grasp the underlying principles and techniques.
Understanding the Expression
The given expression is a product of three terms with the same base (5) and different exponents. The exponents are -3/2, -1/2, and 1/2. To solve this expression, we will follow the exponent rules to simplify it and find its value.
Step-by-Step Solution
The expression can be written as:
5^{-3/2} 5^{-1/2} 5^{1/2}
According to the exponent rules, when you multiply bases with the same base, you can add the exponents:
5^{-3/2} u00d7 5^{-1/2} u00d7 5^{1/2} 5^{-3/2 - 1/2 1/2}
Now, let's simplify the exponent:
-3/2 - 1/2 1/2 -3/2 - 1/2 1/2 -3/2 - 0 -3/2
So, the expression simplifies to:
5^{-3/2}
Now, let's convert the negative exponent to a positive one by taking the reciprocal:
5^{-3/2} 1 / (5^{3/2})
Evaluating the Expression
To further simplify, we know that 5^{3/2} is the same as (5^{1/2})^3, which can be broken down as follows:
5^{3/2} (5^{1/2})^3 (sqrt{5})^3 5 * sqrt{5} / 5 5 * sqrt{5} / 5 (5 * sqrt{5}) / 5
So, the expression becomes:
5^{-3/2} 1 / (5 * sqrt{5}) 1 / (5 * sqrt{5}) * (sqrt{5} / sqrt{5}) (sqrt{5} / (5 * sqrt{5})) / 5 (1 / 5) * sqrt{5}
Final Answer and Approximation
The final simplified form is:
5^{-3/2} (1 / 5) * sqrt{5} 1 / (5 * sqrt{5}) * (sqrt{5} / sqrt{5}) (sqrt{5} / (5 * sqrt{5})) / 5 (1 / 5) * sqrt{5}
Numerically, this is approximately:
2.772724
Thus, the solution to the expression 5^{-3/2} 5^{-1/2} 5^{1/2} is:
(31 sqrt{5}) / 25 2.772724
Conclusion
Understanding and applying exponent rules can greatly help in simplifying and solving mathematical expressions. In this case, we used the product rule for exponents and the reciprocal rule to find the solution. By mastering these techniques, you can handle more complex expressions with ease.
Additional Resources
If you need further assistance or require more examples, please refer to online resources or calculators specifically designed for mathematical expressions.