Problem 60 - Algebra for College Students

Question

Use your calculator to evaluate each numerical expression. $$-4^{9}$$

Step-by-Step Solution

Answer

The value of \\( -4^9 \\\) is \\( -262144 \\\).

Step 1: Understanding the Expression

The expression given is \( -4^9 \). Note that in this case, the negative sign is not included in the exponentiation, as there are no parentheses around -4. Thus, it reads as the negative of 4 to the 9th power.

Step 2: Compute the Power

Calculate \( 4^9 \). Using a calculator, 4 raised to the power of 9 is computed as: \( 4^9 = 262144 \).

Step 3: Apply the Negative Sign

Since the expression is the negative of \( 4^9 \), simply multiply the computed power by -1. Therefore, \( -4^9 = -262144 \).

Key Concepts

Understanding Calculator Usage

When approaching complex mathematical expressions, a calculator is a valuable tool. It simplifies calculations while ensuring accuracy, especially when exponents are involved. For expressions like \(-4^9\), follow these steps on your calculator:

• First, compute the exponent part. Enter the base number (4) and use the power function, often labeled as \(^\wedge\) or **, to raise 4 to the 9th power.
• Next, recognize that the negative sign is not included in the exponent. This means, after calculating the power, simply multiply the result by -1.

Always double-check your entry to ensure correct placement of operations. This habit helps avoid errors during exams or homework sessions.

Handling Negative Numbers in Calculations

Negative numbers can be tricky, especially when they appear with exponents. In the expression \(-4^9\), the negative sign is applied after the exponentiation process.When you encounter such expressions, remember:

• Without parentheses, negative signs come into play after the exponent is calculated. Therefore, the expression reads as the negative result of the power.
• Be mindful of parentheses, as they can change the operation entirely. For example, \((-4)^9\) involves multiplying the base \(-4\) by itself nine times, affecting the overall sign.

Understanding this order can significantly change the outcome, so accuracy and attention to detail are crucial.

Mastering the Order of Operations

The order in which operations are performed can change the result of a calculation. This is often remembered through the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)). Applying this order correctly is crucial.In the expression \(-4^9\):

• First, handle any parentheses. Here, there are none affecting the base and exponent directly.
• Next, address the exponent: calculate \(4^9\).
• Lastly, apply the negative sign. Since it's outside the exponent, it comes last, making it \(-262144\).

Keeping the order of operations organized in your mind ensures that your calculations are correct and efficient.