Understanding and Evaluating Complex Mathematical Expressions: 1/9^{3/2}·1/8^{2/3}

When dealing with complex mathematical expressions, it's essential to understand the order of operations and the principles of exponentiation. This article will guide you through the process of evaluating the expression 1/9^{3/2}·1/8^{2/3} step-by-step. We'll break down each part of the expression and then combine the results for a complete solution.

Evaluating 1/9^{3/2}

The expression 1/9^{3/2} involves both exponentiation and a fraction. According to the order of operations, we first need to evaluate the exponentiation, which is raising 9 to the power of 3/2. Let's break this down:

Step 1: Simplify 9^{3/2}

First, we recognize that 9^{3/2} can be written as {(9^1)^{3/2}}. We know that 9 is 3^2, so we can substitute this in:

9^{3/2} (3^2)^{3/2}

Using the rules of exponents, (a^m)^n a^{m×n}, we get:

(3^2)^{3/2} 3^{2×(3/2)} 3^3

3^3 27. Therefore, 9^{3/2} 27. Now, we need to consider the 1/9 factor. We can rewrite 1/9 as (1/3)^2:

(1/9)^{3/2} ((1/3)^2)^{3/2}

Again, applying the exponent rule (a^m)^n a^{m×n}, we get:

((1/3)^2)^{3/2} (1/3)^{2×(3/2)} (1/3)^3

(1/3)^3 1/27. Thus, 1/9^{3/2} 1/27.

Step 2: Simplify 1/8^{2/3}

Next, we'll evaluate 1/8^{2/3}. Again, according to the order of operations, we first simplify the exponentiation:

Step 2.1: Simplify 8^{2/3}

We recognize that 8 is 2^3, so we can substitute this in:

8^{2/3} (2^3)^{2/3}

Using the exponent rule, we get:

(2^3)^{2/3} 2^{3×(2/3)} 2^2

2^2 4. Thus, 8^{2/3} 4. Now, we need to consider the 1/8 factor. We can rewrite 1/8 as (1/2)^3:

(1/8)^{2/3} ((1/2)^3)^{2/3}

Applying the exponent rule, we get:

((1/2)^3)^{2/3} (1/2)^{3×(2/3)} (1/2)^2

(1/2)^2 1/4. Thus, 1/8^{2/3} 1/4.

Combining the Results

Finally, we need to multiply the results from the two parts together:

(1/9)^{3/2}·(1/8)^{2/3} (1/27)·(1/4) 1/108

So, the value of the given expression 1/9^{3/2}·1/8^{2/3} is 1/108.

Additional Tips for Evaluating Complex Expressions

1. **Order of Operations**: Always follow the order of operations (PEMDAS/BODMAS), which stands for Parentheses/Brackets, Exponents/Orders (powers and roots), Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). 2. **Substitution**: If a number is raised to a fractional exponent, consider rewriting the base number in a different form to make the evaluation easier. 3. **Use of Square Roots**: When you encounter a square root, think about the square of the number inside the root to simplify the expression.

Conclusion

By carefully breaking down the expression and applying the principles of exponentiation, we can evaluate even the most complex mathematical expressions. This example demonstrates the importance of methodical analysis and application of mathematical rules to arrive at the correct solution.