Problem 125 - Algebra A Combined Function

Question

Explain why \((-5)^{4}=625,\) while \(-5^{4}=-625\).

Step-by-Step Solution

Answer

Parentheses include the negative in the exponent's base, affecting the result, while no parentheses means the negative applies after exponentiation.

Step 1: Understanding Exponents and Negative Numbers

When you see an expression like \((-5)^{4}\), it means that the entire value inside the parentheses, \-5\, is raised to the fourth power. This is important because the negative sign is included in the base being multiplied by itself.

Step 2: Calculate \\((-5)^{4}\\)

Compute \((-5)^{4} = (-5) imes (-5) imes (-5) imes (-5)\). Start by multiplying two instances of \-5\: \(-5 imes -5 = 25\). Multiply 25 with another \-5\: \(-5 imes 25 = -125\). Finally, multiply \-125\ by \-5\ to get \(-5 imes -125 = 625\). So, \((-5)^{4} = 625\).

Step 3: Evaluate \\-5^{4}\\ without Parentheses

In the expression \-5^{4}\, exponents take precedence over the negative sign placed before the base. Therefore, \-5^{4}\ is interpreted as \-(5^{4})\. First, calculate \[5^4\]: \5 imes 5 = 25\, \25 imes 5 = 125\, and \125 imes 5 = 625\. Last, apply the negative sign in front, resulting in \-(5^{4}) = -625\.

Key Concepts

Negative Bases

Negative bases can be a bit confusing at first glance, but don't worry! We'll break it down. When we talk about negative bases, we mean expressions where the entire negative number, indicated by parentheses, is raised to a power. Let's look at

• (-5)^{4}: This means the entire number, including the negative sign, is considered the base.
• Every time you multiply, you're multiplying by the negative number itself.

For example, \[(-5)^{4} = (-5) \times (-5) \times (-5) \times (-5)\]
The key here is to remember that multiplying two negative numbers gives you a positive result. So:

• -5 \times -5 = 25, a positive number.
• Now, multiply 25 \times -5 = -125, a negative number.
• Once more, -125 \times -5 = 625, back to a positive result.

This alternation between negative and positive occurs depending on whether the exponent is odd or even. With an even exponent like 4, the negative signs cancel out, resulting in a positive number, 625.

Order of Operations

Order of operations is a fundamental concept in math that dictates the order in which you perform mathematical operations. In expressions with exponents and negative numbers, it's essential to follow these rules, often remembered by the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication
• Division (from left to right)
• Addition
• Subtraction (from left to right)

For the expression \[-5^{4}\], notice the negative sign appears without parentheses. According to the order of operations:

• Compute the exponent first, thus 5^{4}, which equals 625.
• Next, apply the negative sign in front of this result.

So, \[-5^{4} = -(5^{4}) = -625\] By strictly following the order of operations, you ensure each part of the expression plays its correct role, leading to the right result.

Parentheses in Math Expressions

Parentheses are crucial in mathematical expressions because they indicate which operations should be performed first. In terms of precedence in calculations, anything inside parentheses is dealt with before any other operation. When dealing with negative numbers, parentheses can change the entire meaning of an expression.

• If you encounter (-5)^{4}, the parentheses signal that -5 is the base.
• This includes both the negative sign and the 5, so the whole number is raised to the fourth power.

Contrast this with expressions like \[-5^{4}\], where there are no parentheses around -5. Here, only the 5 is raised to the power of 4, which means the entire expression is equivalent to:

• -(5^{4}) = -625.

Understanding how parentheses group parts of an expression helps avoid errors, ensuring that you apply the power to the correct base and account for any negative signs appropriately.