Problem 92 - Precalculus: Functions and Graphs

Question

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 \(\left[(-3)^{15}\right]^{2}\) or \(\left[(-3)^{2}\right]^{1 / 3}\), whereas an error message might otherwise appear. Approximate the real number expression to four decimal places. (a) \((-1.2)^{37}\) (b) \((-5.08)^{7 / 3}\)

Step-by-Step Solution

Answer

(a) -7.7357; (b) -40.7821

Step 1: Understand the Problem

We need to approximate the values of the given expressions \((-1.2)^{37}\) and \((-5.08)^{7/3}\) to four decimal places. These involve raising negative numbers to powers, where fractional powers even often require evaluating roots.

Step 2: Evaluate Expression (a)

For \((-1.2)^{37}\), observe that when a negative number is raised to an odd power, the result is negative. Calculate \((-1.2)^{37}\) using a calculator to approximate: \(-1.2^{37} \approx -7.7357\).

Step 3: Rewrite Expression (b) using Roots

Expression \((-5.08)^{7/3}\) involves a fractional power, which can be split. Rewrite it as:\[((-5.08)^7)^{1/3}\].

Step 4: Calculate Inner Power

Calculate \((-5.08)^7\). Since 7 is an odd integer, the result retains the negative sign:\((-5.08)^7 \approx -68178.4784\).

Step 5: Evaluate the Cube Root

Now, find the cube root of \(-68178.4784\). Use a calculator for approximation:\[-68178.4784^{1/3} \approx -40.7821\].

Step 6: Verify and Approximate

Ensure that each step is calculated with precision, and if necessary, conduct computations with a scientific calculator for accurate results. Double-check your approximations to confirm that they result in \(-7.7357\) for (a) and \(-40.7821\) for (b).

Key Concepts

Negative Numbers

Negative numbers are values that are less than zero. In mathematical computations, they are represented with a minus sign. When dealing with negative numbers, especially when raising them to powers or taking roots, specific rules apply. For example:

• Even powers of negative numbers result in a positive value because the negative signs cancel out.
• Odd powers retain the negative sign, keeping the final result negative.

Understanding these basic properties helps prevent errors in calculations and ensures accurate approximations when solving mathematical problems.

Roots

Roots are essential in understanding fractional powers, especially when they involve negative numbers. The most common roots are square roots and cube roots. Here are some key points about roots:

• Taking the root of a number is the inverse operation of raising a number to a power.
• Square roots of negative numbers are not real numbers and typically involve imaginary units, but odd roots (like cube roots) of negative numbers do exist as real numbers.

When dealing with expressions such as \((-5.08)^{7/3}\), we often rewrite them to separate the power and the root. This method involves first raising the number to the integer power before applying the root.

Odd Powers

Odd powers play a unique role in mathematics, particularly with negative numbers. Here are some details about odd powers:

• When raising a negative number to an odd power, the negative sign is retained.
• Odd powers of numbers can significantly alter their size, either increasing or decreasing them depending on the base and power involved.

For instance, with the expression \((-1.2)^{37}\), since 37 is odd, the outcome remains negative. Odd powers help in understanding how taking a power affects the sign and magnitude of a number.

Real Number Approximation

Complex calculations often require approximations to express results in a more readily understandable form. Here's what to know about approximating real numbers:

• Approximations are necessary when exact calculations involve irrational numbers or impractical lengthy decimals.
• When dealing with fractional powers and roots, calculators are often used to get an approximation accurate to a desired degree, like four decimal places in this example.

Approximating a result, like \(-7.7357\) or \(-40.7821\), helps in understanding the effect of operations such as raising to a power or taking a root, and in making decisions based on these numerical values in real-world applications.