Question

Explain the difference between \((-2)^{4}\) and \(-2^{4}\)

Short answer

Answer: The difference is in the placement of the parentheses and the negative sign. In (-2)^4, the negative sign is included within the exponentiation, resulting in a positive result (16). In -2^4, the negative sign affects the outcome of the exponentiation, leading to a negative result (-16).

Step-by-step solution

Understand the difference between \((-2)^{4}\) and \(-2^{4}\)

Notice that in the first expression, \((-2)^{4}\), the negative sign is inside the parentheses, and in the \(-2^{4}\), the negative sign is outside the parentheses. This difference will affect the result of exponentiation.

Evaluate \((-2)^{4}\)

In this expression, the entire number, including the negative sign, is raised to the power of 4. Since \((-2) \times (-2) = 4\) and \((4) \times (4) = 16\), we can evaluate the expression as: $$ (-2)^{4} = (-2) \times (-2) \times (-2) \times (-2) = 4 \times 4 = 16 $$

Evaluate \(-2^{4}\)

In this expression, only the number 2 is raised to the power of 4, and then the negative sign is applied. Since \(2^{4} = 2 \times 2 \times 2 \times 2 = 16\), we can evaluate the expression as: $$ -2^{4} = -(2^{4}) = -(2 \times 2 \times 2 \times 2) = -16 $$

Compare the results and explain the difference

When we compare the results, we can see that \((-2)^{4} = 16\) while \(-2^{4} = -16\). The difference comes from the placement of the parentheses and the negative sign. In the first expression, the negative sign is included within the exponentiation, resulting in a positive result. In the second expression, the negative sign affects the outcome of the exponentiation, leading to a negative result.

Key concepts

Exponentiation

Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The exponent tells us how many times to multiply the base by itself. For example, in the expression \(2^4\), the base is 2, and the exponent is 4. This means we multiply 2 by itself four times: \(2 \times 2 \times 2 \times 2 = 16\).

When working with exponents, always perform them before any addition, subtraction, multiplication, or division, following the order of operations (PEMDAS/BODMAS). The placement of parentheses, as we will see, has a significant impact on the result of exponentiation.

Negative Numbers

Negative numbers are values less than zero, represented with a minus sign (\(-\)). When dealing with negative bases and exponents, it's crucial to understand how negative signs interact with mathematical operations.

For example:

• Raising a negative number to an even power results in a positive number. Such as \((-2)^4\), which equals 16 because the product of two negative numbers is positive.
• Raising a negative number to an odd power results in a negative number. Such as \((-2)^3\), which equals \(-8\) because one negative pair remains unattended.

In expressions like \(-2^4\), where the negative sign is not in parentheses with the base, it simply applies after exponentiation, resulting in \(-16\). This difference can be puzzling, making it important to carefully read each part of the expression.

Parentheses Usage

Parentheses play a vital role in mathematical expressions by clearly defining which parts of an expression to evaluate first.

In our example:

• \((-2)^4\): Parentheses include the negative sign with the base, indicating that the entire number, \(-2\), should be raised to the fourth power.
• \(-2^4\): Here, the negative sign is outside the parentheses, so only 2 is raised to the power of 4, and then the negative sign is applied to the result.

Using parentheses accurately ensures the correct execution of operations and helps in avoiding errors that may lead to incorrect results.

Mathematical Expressions

Mathematical expressions involve various operations like addition, subtraction, multiplication, division, and exponentiation.

Understanding these expressions requires carefully analyzing each component and following the order of operations (PEMDAS/BODMAS).

Here’s how you should approach expressions:

• Identify all operators and their priorities.
• Look for parentheses to determine the sequence of operations.
• Evaluate exponents early in solving.
• Perform multiplication and division from left to right.
• Lastly, handle addition and subtraction from left to right.

In cases involving both parentheses and exponents, as in our original exercise, recognizing how they interact is crucial for accurate solutions.