Problem 6
Question
Evaluate each exponential expression. $$ -2^{4} $$
Step-by-Step Solution
Verified Answer
The value of \(-2^4\) is 16.
1Step 1: Understanding the Exponent
The symbol '^' represents 'raised to the power of'. So, \(-2^4\) means -2 is raised to the power of 4, which means that -2 is multiplied by itself 4 times.
2Step 2: Multiplying -2 four times
If -2 is multiplied by itself 4 times, it becomes: \((-2) \times (-2) \times (-2) \times (-2)\). Because a negative number multiplied by a negative number results in a positive number, every two negative factors will yield a positive product.
3Step 3: Evaluate the Expression
Calculate the multiplication: \((-2) \times (-2) = 4\) and \(4 \times 4 = 16\). Thus, \(-2^4\) equals to 16.
Key Concepts
Understanding ExponentsHandling Negative NumbersOrder of Operations
Understanding Exponents
Exponents can be thought of as shorthand for repeated multiplication. For example, when you see something like \(2^3\), it simply means that you should multiply 2 by itself three times: \(2 \times 2 \times 2 = 8\).
Exponents have certain features:
Exponents have certain features:
- Base Number: The number being multiplied, like the 2 in \(2^3\).
- Exponent: Indicates how many times to multiply the base by itself. In \(2^3\), the 3 is the exponent.
Handling Negative Numbers
Negative numbers can change the results of calculations significantly, especially in exponents. When you multiply two negative numbers, the result is positive. This simple rule applies even when you multiply more than two numbers together.
For instance, \((-2) \times (-2) = 4\). Here's where it gets interesting: if you continue multiplying, and you have an odd number of negative factors, the result will be negative. However, with an even number of negative factors, your result remains positive.
The expression \((-2)^4\) becomes positive 16 because the four negative signs "cancel" each other out in pairs, leaving a positive number. But keep in mind that it differs from \(-2^4\) without the parentheses, which first squares the base, then applies the negative sign, resulting in -16.
For instance, \((-2) \times (-2) = 4\). Here's where it gets interesting: if you continue multiplying, and you have an odd number of negative factors, the result will be negative. However, with an even number of negative factors, your result remains positive.
The expression \((-2)^4\) becomes positive 16 because the four negative signs "cancel" each other out in pairs, leaving a positive number. But keep in mind that it differs from \(-2^4\) without the parentheses, which first squares the base, then applies the negative sign, resulting in -16.
Order of Operations
The order of operations is a fundamental principle in mathematics that ensures consistency in calculations. Sometimes remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)), it guides you on which operations to perform first in a mathematical expression.
In our example \(-2^4\), following the order of operations correctly means handling the exponent first. This requires you to calculate \(2^4 = 16\) before applying the negative sign. This is because the exponent only applies to the number directly before it and not to the entire expression unless parentheses are used. Thus, the correct evaluation is \(-2^4 = -(16) = -16\).
Always remember, neglecting the order of operations can lead to incorrect results, especially when dealing with more complex expressions.
In our example \(-2^4\), following the order of operations correctly means handling the exponent first. This requires you to calculate \(2^4 = 16\) before applying the negative sign. This is because the exponent only applies to the number directly before it and not to the entire expression unless parentheses are used. Thus, the correct evaluation is \(-2^4 = -(16) = -16\).
Always remember, neglecting the order of operations can lead to incorrect results, especially when dealing with more complex expressions.
Other exercises in this chapter
Problem 6
In Exercises \(1-10\), factor out the greatest common factor. $$6 x^{4}-18 x^{3}+12 x^{2}$$
View solution Problem 6
Evaluate each expression in Exercises \(1-6\) or indicate that the root is not a real number. $$\sqrt{(-17)^{2}}$$
View solution Problem 6
$$\text {find the degree of the polynomial.}$$ $$-4 x^{3}+7 x^{2}-11$$
View solution Problem 7
Simplify each rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. $$ \frac{3 x-9}{x^{2}-6 x+9} $$
View solution