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Chapter 6: Problem 170

Explain why \(-5^{3}=(-5)^{3}\) but \(-5^{4} \neq(-5)^{4}\).

Short Answer

Expert verified
-5^{3} = (-5)^{3} because both expressions result in -125. -5^{4} eq (-5)^{4} because the first results in -625, while the second results in 625.

Step by step solution

01

Understand the Expression \( -5^{3} \)

Here, \( -5^{3} \) means that we are taking the cube of 5 and then multiplying it by -1. This is because the negative sign is not inside the parenthesis.So, \[ -5^{3} = -(5^{3}) = -125 \].
02

Understand the Expression \( (-5)^{3} \)

Here, \( (-5)^{3} \) means that we are taking the cube of -5. Since the negative sign is inside the parenthesis, it is included in the exponentiation.So, \[ (-5)^{3} = (-5) \times (-5) \times (-5) = -125 \].
03

Understand the Expression \( -5^{4} \)

Here, \( -5^{4} \) means that we are taking the fourth power of 5 and then multiplying it by -1. This is because the negative sign is not inside the parenthesis.So, \[ -5^{4} = -(5^{4}) = -625 \].
04

Understand the Expression \( (-5)^{4} \)

Here, \( (-5)^{4} \) means that we are taking the fourth power of -5. Since the negative sign is inside the parenthesis, it is included in the exponentiation.So, \[ (-5)^{4} = (-5) \times (-5) \times (-5) \times (-5) = 625 \].
05

Comparing the Results

From the calculations:* \( -5^{3} = -125 \)* \( (-5)^{3} = -125 \)* \( -5^{4} = -625 \)* \( (-5)^{4} = 625 \)We see that \( -5^{3} = (-5)^{3} \) but \( -5^{4} eq (-5)^{4} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

powers of negative numbers
Negative numbers can be tricky, especially when dealing with exponents. When you raise a negative number to an exponent, the result depends on the parity (odd or even) of the exponent.
For odd exponents, the result will be negative. For example, \((-5)^3 = -125\).
For even exponents, the result will be positive. For instance, \((-5)^4 = 625\).
This is because multiplying an odd number of negative factors together results in a negative product, while multiplying an even number of negative factors gives a positive product.
parentheses in exponents
Parentheses play a crucial role in mathematical expressions involving exponents. They help clarify which part of the expression is affected by the exponent.
For instance, \(-5^3\) means that the exponent only applies to 5, and then you multiply by -1 afterwards. This gives you \(-5^3 = -(5^3) = -125\).
On the other hand, \((-5)^3\) means that the entire negative number -5 is raised to the power, resulting in \((-5)^3 = (-5) \times (-5) \times (-5) = -125\).
As is evident, without parentheses, the negative sign is not part of the base being raised to the power.
exponent rules
Understanding exponent rules is key to mastering operations with exponents. Here are some fundamental rules:
  • Product of Powers Rule: \(a^m \times a^n = a^{m+n}\)
  • Power of a Power Rule: \((a^m)^n = a^{mn}\)
  • Power of a Product Rule: \((ab)^m = a^m \times b^m\)
  • Power of a Quotient Rule: \((a/b)^m = a^m / b^m\)

These rules help simplify and solve exponent-related problems. Notably, when the base is negative and inside parentheses, both the negative sign and the base are raised to the power. If the negative sign is outside the parentheses or without them, it is applied after exponentiation.

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