Understanding Negative Exponents and Bases in Algebraic Expressions

When you keep the number key pressed down, it's as if you're typing in (-2)^6. The calculator will calculate (-2)^5. However, once you release the key, it will take a negative of the result and give -32.In summary, the conversation discusses the difference between ##-r^4## and ##(-r)^4## and how this affects the final result in an expression. The conversation also addresses the use of parentheses and the order of operations when simplifying expressions. Finally, the calculator's method of operation is mentioned as a potential source of confusion.
  • #1
DS2C
Going through a problem and and I keep getting it wrong and I'm not sure why.
In a part of the problem, the expression ##\left(-3\right)\left(-r^4\right)\left(-s^5\right)## comes up and the solution that it's giving me is ##-3r^4s^5##
Wouldn't the last factor be ##-s^5## since the power of a base with an odd exponent should be negative? Not sure if I'm tripped up somewhere but the book specifically states this, and then gives this solution.
 
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  • #2
DS2C said:
Going through a problem and and I keep getting it wrong and I'm not sure why.
In a part of the problem, the expression ##\left(-3\right)\left(-r^4\right)\left(-s^5\right)## comes up and the solution that it's giving me is ##-3r^4s^5##
Wouldn't the last factor be ##-s^5## since the power of a base with an odd exponent should be negative? Not sure if I'm tripped up somewhere but the book specifically states this, and then gives this solution.
There's a difference between ##-r^4## and ##(-r)^4## that you seem to be overlooking. The latter equals ##r^4##, which is the opposite sign of ##-r^4##. In the second and third factors, the bases are, respectively, r and s, not (-r) and (-s).
##(-3)(-r^4)(-s^5) = (-1)^3 \cdot 3r^4s^5 = -3r^4s^5##.
 
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  • #3
To my understanding, ##-r^4## simply means "the negative of ##r^4##.
So ##-r^4=-\left(r\right)\left(r\right)\left(r\right)\left(r\right)##, and it will always be a negative no matter how many r factors there are and no matter if there is an even or odd amount.
Similarly, ##\left(-r\right)^4=\left(-r\right)\left(-r\right)\left(-r\right)\left(-r\right)=r^4##
However, if it were ##\left(-r\right)^5##, would this not be a negative since there is an odd exponent? Or am I just getting hung up on the even/odd exponent ordeal and way overthinking it? Will ##\left(-r\right)^4## and ##\left(-r\right)^5## BOTH have positive results?
For clarification, I plugged ##\left(-2\right)^5## into my calculator and it gave me -32. I then plugged in ##\left(-2\right)^6## and it gave me +64. So according to that train of thought, if the exponent is odd then the result will be negative and if the exponent is even then the result will be positive.
Kind of a dumb question, thank you for taking the time.
 
  • #4
There is no (-r)5 involved.

Your expression is ##\left(-3\right)\left(-r^4\right)\left(-s^5\right) = (-3)(-(r^4))(-(s^5)) = (-3)(-1)(r^4)(-1)(s^5)## with additional brackets added to make the association clearer. If you simplify the last expression, you'll see that the exponents of r and s are irrelevant here.
 
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  • #5
Ok I think I understand. I just used the ##\left(-r\right)^5## as an example not pulled from the actual expression.
I understand what youre both saying. But why does the calculator come up with something different, as in my last post?
 
  • #6
DS2C said:
For clarification, I plugged ##\left(-2\right)^5## into my calculator and it gave me -32. I then plugged in ##\left(-2\right)^6## and it gave me +64. So according to that train of thought, if the exponent is odd then the result will be negative and if the exponent is even then the result will be positive.
##(-2)^5## is the same as ##(-1)^5(2)^5##, which is -32. For the other expression, you have ##(-1)^6## times ##2^6##, or 64.
 
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  • #7
Ok thanks for the help everyone.
 
  • #8
The calculator probably operates step by step. You type in -2, then ^5, and it calculates (-2)^5.
 

1. What is a negative base with an exponent?

A negative base with an exponent is a mathematical expression where a negative number is raised to a power. This means that the number is multiplied by itself a certain number of times, determined by the exponent.

2. How do you evaluate a negative base with an exponent?

To evaluate a negative base with an exponent, you can use the rule that states that a negative number raised to an even exponent will result in a positive number, while a negative number raised to an odd exponent will result in a negative number.

3. Can a negative base with an exponent be simplified?

Yes, a negative base with an exponent can be simplified by using the rules of exponents. For example, if the exponent is a fraction, the negative base can be rewritten as a fraction with a positive base and the exponent as the numerator.

4. What are some real-life applications of negative bases with exponents?

Negative bases with exponents are commonly used in scientific calculations, such as in physics and chemistry. They can also be used in financial calculations, such as calculating compound interest.

5. Are there any limitations to using negative bases with exponents?

One limitation of using negative bases with exponents is that they can sometimes result in complex numbers. This can make it difficult to interpret the result and may require additional steps to simplify the expression. Additionally, some calculators may not be able to handle negative bases with exponents, so manual calculations may be necessary.

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