Problem 58 - Algebra for College Students

Question

Use your calculator to evaluate each numerical expression. $$(-2)^{8}$$

Step-by-Step Solution

Answer

The result is 256.

Step 1: Understanding the Expression

The expression we need to evaluate is \((-2)^8\). This means we need to multiply -2 by itself 8 times.

Step 2: Multiply Repeatedly

Calculate the repeated multiplication:1. \((-2) \times (-2) = 4\)2. \(4 \times (-2) = -8\)3. \(-8 \times (-2) = 16\)4. \(16 \times (-2) = -32\)5. \(-32 \times (-2) = 64\)6. \(64 \times (-2) = -128\)7. \(-128 \times (-2) = 256\)So, \((-2)^8 = 256\).

Step 3: Use a Calculator

Alternatively, you can input \((-2)^8\) into a scientific calculator. Ensure that the base, -2, is inside parentheses to correctly evaluate the expression with the negative sign. The calculator should give you the result 256.

Step 4: Verify the Result

The result of the calculation \((-2)^{8}\) is 256. This result is consistent with both manual calculations and calculator output.

Key Concepts

Calculator Usage

Using a calculator to evaluate exponential expressions like \((-2)^8\) is straightforward if you know how to input the expression correctly. It's important to use parentheses around the base number when it is negative. Doing so ensures that the negative sign is included in the repeated multiplication process. To evaluate \((-2)^8\), follow these steps:

• Turn on the calculator and ensure it is in the right mode (usually default is fine for basic exponents).
• Type the negative base as \((-2)\). Use the parentheses to preserve the negative sign.
• Press the exponentiation button which typically looks like a caret (^).
• Enter the exponent \(8\).
• Press "enter" or "equals" to see the result, which should be 256.

Using calculators for such expressions not only saves time but can also reduce human error. Always double-check that you've inputted the expression exactly as it appears in the problem, particularly paying attention to any signs or parentheses.

Negative Exponents

Negative exponents might seem confusing at first, but they follow a simple rule: a negative exponent indicates the reciprocal of the base raised to the opposite positive exponent. For instance, \(a^{-n} = \frac{1}{a^n}\).However, in the expression \((-2)^8\), we have a negative base but a positive exponent, which involves straightforward multiplication without needing to worry about reciprocals or fractions. Here’s what happens with negative bases:

• A negative base raised to an even exponent results in a positive number because all pairs of negative factors multiply to give a positive product.
• A negative base raised to an odd exponent results in a negative number because there’s an unpaired negative factor influencing the sign.

In our case of \((-2)^8\), the exponent is even, hence multiplying \((-2)\) by itself 8 times results in a positive number, which is 256.Understanding these properties helps in recognizing patterns and predicting results without extensive calculations, especially when working with larger negative bases and exponents.

Order of Operations

The order of operations is a fundamental principle that ensures calculations are done correctly and consistently. The standard order can be remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)).When dealing with exponential expressions like \((-2)^8\), knowing the order of operations is crucial for correctly calculating the power:

• Parentheses: Evaluate anything inside parentheses first, which in our expression is just the base value \((-2)\).
• Exponents: Next comes exponents; calculate the power by multiplying the base by itself the number of times specified by the exponent.

For our expression \((-2)^8\), the order dictates that we handle the exponentiation right after considering any mathematical expressions inside the parentheses. It is essential because if parentheses were ignored, one might mistakenly compute the exponent first and then apply the negative sign, resulting in an incorrect answer.By strictly adhering to the order of operations, you ensure accurate results across different mathematical problems and avoid common mistakes.