Problem 82 - College Algebra and Calculus: An Applied Approach

Question

Calculator Write the keystrokes you can use to evaluate \(\sqrt[3]{(-5)^{5}}\) in one step on your calculator.

Step-by-Step Solution

Answer

The correct sequence of keystrokes on the calculator is ( ( -5 ) ^ 5 ) ^ (1/3).

Step 1: Identify the base and power

The base is -5 and the exponent is 5. In most calculators, the power function is represented as '^'. Inputting -5^5 will cause the calculator to calculate -5 raised to the 5th power. Note that the negative sign must be included inside parentheses to correctly calculate the negative base to a power.

Step 2: Apply the root function

The calculating expression should now be evaluated as the cube root. This is typically represented as \( \sqrt[3]{} \). Some calculator might use the '^' operator followed by the fraction 1/3 to represent the cube root operation. Therefore, following the power calculation, the whole expression is wrapped in parentheses and the cube root operation is applied outside the parentheses. This results in \( \sqrt[3]{(-5)^{5}} \) being accurately evaluated.

Key Concepts

Cubic Root Calculation

Calculating the cubic root, also known as the cube root, is a fundamental operation in mathematics. When you find the cubic root of a number, you're essentially asking, "What number, when multiplied by itself twice, gives the original number?" For example, the cube root of 27 is 3, because 3 * 3 * 3 equals 27. To calculate the cubic root on a calculator, you often use specific buttons or functions:

• Some calculators have a dedicated cubic root button, which makes the process simple.
• Other calculators might require using the exponentiation function with a fraction, such as raising the number to the power of \(\frac{1}{3}\).

When dealing with expressions like \(\sqrt[3]{(-5)^{5}}\), ensure you find \((-5)^{5}\) first and then take the cube root of the result. Understanding how to calculate cubic roots helps in various applications, including algebra and geometry.

Exponentiation

Exponentiation is the mathematical operation involving two numbers, the base and the exponent. In this operation, the base is repeatedly multiplied by itself as many times as the exponent indicates. For instance, 5 raised to the power of 3, written as \(5^3\), equals 5 * 5 * 5, which is 125. When dealing with a negative base, as in -5 raised to the power of 5, the calculation becomes slightly more intricate. Parentheses are crucial here.

Importance of Parentheses

Using parentheses ensures the entire negative number is raised to the power, not just the numeric part:

• \((-5)^5\) results in a negative number because the base (-5) is multiplied by itself an odd number of times, maintaining the sign.
• In contrast, \(-5^5\) can mistakenly be calculated by a calculator as -(5 * 5 * 5 * 5 * 5), making the negative apply only after exponentiation, leading to incorrect results.

Make sure to use your calculator to its full potential by understanding how to accurately input and process negative bases with exponents.

Negative Base Handling

Handling negative bases in calculations, especially with exponentiation, requires careful attention to detail. When raising a negative number to a power, the rules of exponents dictate specific outcomes:

• An odd exponent retains the negative sign. For example, \((-5)^3 = -125\).
• An even exponent changes the result to a positive number due to the negative sign being multiplied evenly, such as \((-5)^2 = 25\).

It's also essential to consistently use parentheses, ensuring the negative base is taken into account correctly in operations:1. Always enclose the entire negative base in parentheses.2. When entered into a calculator, this signals the device to handle the entire term, not just the numerical part, in the operation.Getting the parentheses and inputs right can prevent miscalculations and lead to the correct results when dealing with negative numbers, particularly in advanced mathematical expressions involving roots and powers.