Question

Without a calculator, decide whether the quantities are positive or negative. $$ -3^{4} $$

Short answer

Answer: The result of the expression $$-3^4$$ is negative.

Step-by-step solution

Analyze expression and recall order of operations

The given expression is $$-3^4$$. Before solving, let's recall the order of operations: Parentheses, Exponents, Multiplication/Division (from left to right), and Addition/Subtraction (from left to right).

Apply the exponent first

According to the order of operations, we handle the exponent first. We have $$3^4 = 3\times 3\times 3\times 3 = 81$$.

Apply the negative sign

Now that the exponent has been evaluated, we apply the negative sign while keeping in mind that it doesn't affect the base of the exponent: $$-3^4 = -81$$.

Determine whether the result is positive or negative

Since the result is $$-81$$, the given quantity is negative.

Key concepts

Exponents

Exponents represent the concept of repeated multiplication of a number by itself. In the expression \(3^4\), the number 3 is the base, and the number 4 is the exponent. This means you multiply 3 by itself 4 times:

• 3 multiplied by 3 equals 9.
• 9 multiplied by 3 equals 27.
• Finally, 27 multiplied by 3 equals 81.

So, \(3^4 = 81\). When dealing with exponents, it's essential to perform this operation first according to the order of operations. This explains why in the original exercise, we solved the exponent before applying any other operations.

Negative Numbers

Negative numbers are less than zero and are represented by a minus sign (-). In the context of the operation \(-3^4\), it's crucial to understand where the negative sign applies. The expression means that we first calculate the exponent as per the order of operations, and then apply the negative sign.
The negative sign can change the result from positive to negative if it's placed in front of a non-negative number. Here, the calculation gives us \(3^4 = 81\), and then applying the negative sign results in \(-81\).
Remember, the negative sign is not part of the base in this instance, which is a common mistake. Always deal with exponents first, then apply the negative sign outside of the exponent.

Algebraic Expressions

Algebraic expressions can include numbers, variables, exponents, and operations. In an expression like \(-3^4\), variables aren't present, but the concepts still apply. Here, understanding is based on the components of numbers, negative signs, and exponents.

• The order of operations (PEMDAS/BODMAS) helps decide which part of the expression to solve first.
• We treat \(3^4\) as one part to evaluate first, and then apply the negative sign.
• This method ensures that complex expressions are simplified in a manageable and consistent way.

Algebraic expressions are about recognizing these parts and applying arithmetic rules properly, so it's important to focus on how parts of the equation interact. Even simple-seeming expressions, like \(-3^4\), demand careful application of operations to avoid mistakes.