Simplifying and Solving Complex Exponential Expressions

When working with complex exponential expressions, understanding and applying basic algebra can significantly simplify the process. This guide will demonstrate how to solve an expression involving different fractional exponents and provide a step-by-step approach to tackle such problems. We will also highlight the importance of approximate values in certain cases.

Understanding the Problem

Consider the expression: 51/3 times; 52/3. The initial goal is to simplify this expression step by step using algebraic manipulation.

Simplifying the Expression

Start by introducing a substitution. Let x 51/3. This substitution simplifies the expression:

With these substitutions, the original expression can be rewritten as:

51/3 times; 52/3 x3 times; x2 times; x

Next, combine the terms:

x3 times; x2 times; x x3 2 1 x6

Now, substitute back to the original variable:

56 15625

Using the Quadratic Formula

Another approach involves factoring and using the quadratic formula. Start with the expression x3 - x - 1:

Identify the quadratic term, which is x2 - x - 1.

Use the quadratic formula to find the roots of the quadratic expression. The formula is:

x u00BD(-b ± sqrt(b2 - 4ac))

Substitute a 1, b -1, and c -1 into the formula:

x u00BD(1 ± sqrt(1 - 4 times; (-1)))

x u00BD(1 ± sqrt(5))

Solve for the roots:

x u00BD(1 ± 2.236)

x 1.618 (approx) or x -0.618 (approx)

Since the quadratic expression does not have real roots, the expression is always positive for real values.

Approximate Value Calculation

The exact value of the expression is 52/3 times; 51/3, but an approximate value can be calculated directly using a calculator:

51/3 ≈ 1.710

52/3 ≈ 2.924

Summing these values:

5 51/3 52/3 ≈ 5 1.710 2.924 ≈ 9.634

Conclusion

The expression 51/3 times; 52/3 simplifies to 51 5. However, if we need an approximate value, we can calculate it as 9.634.

Additional Considerations

When faced with similar expressions, the key steps are to introduce substitutions, simplify the expression, and determine whether a more complex method like the quadratic formula is necessary. Approximate values often provide practical solutions for real-world applications.