Problem 25 - Prealgebra

Question

Use the definition of exponents to expand each of the following expressions. Then multiply according to the rule for multiplication. a. \((-2)^{4}\) b. \(-2^{4}\)

Step-by-Step Solution

Answer

a. \((-2)^4 = 16\); b. \(-2^4 = -16\).

Step 1: Understanding Exponents

The expression \((-2)^{4}\) means that \(-2\) is to be multiplied by itself four times: \((-2)\times(-2)\times(-2)\times(-2)\). Similarly, \(-2^{4}\) means we apply the exponent only to \(2\), giving us: \(-(2\times 2\times 2\times 2)\).

Step 2: Expand and Simplify \((-2)^{4}\)

Since \((-2)\times(-2)\times(-2)\times(-2)\) involves four multiplications of negative numbers, multiplication in pairs will give: \((-2)\times(-2) = 4,\)\( (4)\times(-2) = -8,\)\( (-8)\times(-2) = 16.\) Thus, \((-2)^{4} = 16\).

Step 3: Expand and Simplify \(-2^{4}\)

For \(-2^{4}\), the \(2^{4}\) is calculated first: \(2\times 2\times 2\times 2 = 16\) and then multiplied by \(-1\). Thus, \(-2^{4} = -16\).

Key Concepts

Multiplication Rules

When dealing with exponents, multiplication rules become very handy. Multiplication is one of the essential operations in math, and when combined with exponents, certain properties make calculations easier. When multiplying similar bases with exponents, we add their exponents. For instance, multiplying \(x^a \times x^b\) results in \(x^{a+b}\). Notice how the bases stay the same and only the exponents are altered.

• For example, if we have \(3^2 \times 3^3\), it becomes \(3^{2+3} = 3^5\).
• This rule simplifies operations as it allows us to transform expanded multiplication of exponents into a single term.

In the original exercise, we are given expressions like \((-2)^4\), which means multiplying \(-2\) by itself four times. By expanding it, \((-2) \times (-2) \times (-2) \times (-2)\), you can systematically multiply in pairs, making the calculation straightforward and utilizing multiplication rules effectively.

Negative Exponents

Negative exponents can initially seem confusing, but they actually follow a simple logic. A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent. This means \(a^{-n} = \frac{1}{a^n}\). Negative exponents are a concise way of expressing division.

• For example, \(5^{-2} = \frac{1}{5^2} = \frac{1}{25}\).
• Similarly, \(x^{-3} = \frac{1}{x^3}\) expresses that \(x\) is dividing rather than multiplying.

In the exercise, handling \(-2^4\) means understanding that \(-1\) is outside the exponent, applying multiplication rules after calculating the power of 2. However, if it were \((-2)^4\), it leads to a positive result, as multiplying negative values an even number of times turns them positive. Recognizing where the negative sign is placed in the exponentiation makes a substantial difference in the final calculation.

Order of Operations

The order of operations dictates the correct sequence to solve math problems, avoiding common mistakes. It's often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction).

• Parentheses signal operations that must be done first.
• Exponents come next, making it essential to handle them right after parentheses.
• Multiplication and division follow, executed from left to right as they appear in the expression.
• Lastly, addition and subtraction are completed, also from left to right.

In our exercise with \((-2)^4\), parentheses indicate an adjustment of priority, specifying that \(-2\) should be considered as a unit before applying the exponent. Without altering the order explicitly, as in \(-2^4\), we compute \(2^4\) first, then multiply the result by \(-1\). Understanding and following these steps correctly ensures accuracy in calculations and the accurate application of exponents.