Problem 59 - Intermediate Algebra

Question

Use your calculator to evaluate each numerical expression. $$ (-2)^{11} $$

Step-by-Step Solution

Answer

The value of \((-2)^{11}\) is -2048.

Step 1: Understand the Expression

The expression given is \((-2)^{11}\). This means we are raising -2 to the power of 11.

Step 2: Use a Calculator

Enter the base \(-2\) into the calculator and then use the power/exponent button to input 11. Calculate the result.

Step 3: Evaluate the Power

The calculator will compute \((-2)^{11}\) to give you the final result. Pay attention to the negative sign because raising a negative number to an odd exponent results in a negative number.

Key Concepts

Negative Numbers

Negative numbers are less than zero and represented with a minus sign (-) in front of the number. They are important to understand when dealing with mathematical operations like addition, subtraction, multiplication, and division, because they behave differently than positive numbers.

When raising negative numbers to powers, special rules apply. If a negative number is raised to an odd power, the result is negative. For example, \((-2)^3 = (-2) imes (-2) imes (-2) = -8\).

• This happens because multiplying two negative numbers results in a positive number, but adding another negative number turns the result negative again.

On the other hand, if the negative number is raised to an even power, the result is positive, such as \((-2)^2 = (-2) imes (-2) = 4\).

Understanding these rules helps in evaluating expressions correctly and avoids errors when working with different powers.

Powers of Numbers

The concept of 'powers of numbers', also known as exponentiation, involves raising a number (the base) to a power (the exponent or index). This means multiplying the base by itself as many times as the exponent indicates. It is a way to express repeated multiplication succinctly.

For example, if we have a base number \(2\) and an exponent of \(3\), the expression \(2^3\) equals \(2 \times 2 \times 2\), which results in \(8\).

• When working with powers, it is essential to differentiate whether the power is applied to just the number or to the number including a negative sign, as in \((-2)^3\).
• Remember, exponentiation has precedence over multiplication and addition in the order of operations (PEMDAS/BODMAS).

Understanding the rules of powers helps in solving algebraic expressions and equations effectively.

Calculator Usage

Calculators are useful tools for quickly evaluating numerical expressions involving complex operations like exponentiation. Knowing how to input expressions correctly in a calculator is just as important as solving them manually.

When using a calculator to raise a number to a power:

• Enter the base number first. For example, if you're evaluating \((-2)^{11}\), input \(-2\).
• Find and use the exponentiation button. This button could look like \(^\), or it may be a dedicated "power" button on your calculator.
• Enter the exponent. For \((-2)^{11}\), input \(11\) after using the exponentiation button.

Completing these steps correctly should give you the result of the operation. Pay particular attention to using parentheses where needed, especially with negative numbers, to ensure that the calculator calculates the power properly. This avoids mistakes and ensures accurate results.