Problem 91 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 91-92: In evaluating negative numbers raised to fractional powers, it may be necessary to evaluate the root and integer power separately. For example, \((-3)^{2 / 5}\) can be evaluated successfully as \(\left[(-3)^{1 / 5}\right]^{2}\) or \(\left[(-3)^{2}\right]^{1 / 5}\), whereas an error message might otherwise appear. Approximate the realnumber expression to four decimal places. $$ \text { (a) }(-3)^{2 / 5} $$

Step-by-Step Solution

Answer

The approximate value of \\((-3)^{2/5}\\) is 1.5518.

Step 1: Understanding the Expression

The expression \((-3)^{2/5}\) involves a fractional exponent, which can be interpreted as a root followed by a power. Here, \((-3)^{2/5} = \left[(-3)^{1/5}\right]^2\), which means we first find the fifth root of \-3, then square the result.

Step 2: Dealing with Negative Base

Since we are taking the fifth root of a negative number, we are looking for a value where \((a^5 = -3)\). Real roots (odd roots) for negative numbers are negative, so we can find the fifth root and then proceed to the next step.

Step 3: Calculating the Fifth Root

We calculate \((-3)^{1/5}\). Use a calculator to find this value, which is approximately \-1.2457. This gives us the fifth root of \-3.

Step 4: Calculating the Square of the Root

Next, square the result of the fifth root. Compute \((-1.2457)^2\), which results in approximately \1.5518\.

Step 5: Verifying Calculations

Double-check your calculations to ensure no mistakes were made in evaluating the expressions through successive operations (fifth root, then square). This guarantees the accuracy of the approximation.

Key Concepts

Evaluating Negative Numbers

When dealing with negative numbers in mathematical expressions, especially those with fractional exponents, it's essential to approach them carefully. In the expression \((-3)^{2/5}\), we notice that the base, -3, is negative. This indicates that the standard rules of exponents must be applied with additional considerations.

• Negative bases can create particular cases, especially when combined with fractional exponents.
• Odd roots of negative numbers are real and negative, while even roots may involve imaginary numbers.

Understanding these peculiarities helps in handling computations effectively and ensuring accurate results.

Root and Power Separately

Fractional exponents can often be split into a root operation followed by a power. This approach simplifies calculations and avoids errors, especially with negative bases. For example, consider \((-3)^{2/5}\). Splitting the exponent

• Read \(2/5\) as performing a fifth root first, then squaring the result.
• This translates to \( \left[(-3)^{1/5}\right]^2\).

This separation is critical:This separation especially avoids input errors on calculators, because they can sometimes mishandle fractional powers with negative bases. This method emphasizes completing one operation at a time, ensuring precision at each step.

Real Number Approximation

Once operations are defined and calculated separately, approximating real values to a certain number of decimal places provides an accurate and usable result. Calculating \((-3)^{1/5}\) using a calculator yields approximately \(-1.2457\). Real approximation is all about precision:

• Use a reliable calculator or computational tool to find values of roots or powers.
• Ensure results are rounded correctly, typically to four decimal places, for consistency in answers.

This step is crucial for applications in mathematics, science, or engineering, where accurate numeric expressions are needed.

Negative Base Exponents

Fractions as exponents create a dual operation of roots and powers applied to the base. When the base is negative, understanding how negative values behave under different mathematical operations is vital.

• With odd denominators in the fraction of the exponent, such as fifths, roots of negative bases are negative.
• This example shows how \((-3)^{2/5}\) uses negative base properties to achieve a real solution.

Negative base exponents require careful handling during calculation, to ensure correct application of rules and deriving the expected result. It's like unraveling a process where each decision impacts the final approximation.