Exponents clarification question

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In summary, you have an equation with a = symbol that has a different numerator and denominator then what is given.
  • #1
hackedagainanda
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Homework Statement
Simplify and remove all negative exponents from this equation:

## \frac {2k^2 k^3} {k^{-1} k^{-5}}(5k^{-2})^{-3}##
Relevant Equations
Quotient Rule: ##\frac {x^a} {x^b} = x^{a-b}##

Product Rule: ## x^a x^b = x^{a+b}##
So my attempt is this: ##(2k)^2 k^3 = 4k^5## to clear the top numerator then to clear the denominator ## k^{-1} k^{-5} = k^{-6}##Then I apply the quotient rule and get ##4k^{11} (5k^{-2})^{-3}## and simplifying the right hand side I get ##5^{-3} k^6## here is where I got lost, why is it that when you use a negative exponent on the 5 to get 1/125 why do you apply the ##k^{17}## to numerator and not the denominator ##\frac {4k^{17}} {125}## Why isn't the answer ## \frac {4} {125k^{17}}## ?

I probably forgot the rules from arithmetic most likely, I feel embarrassed to ask but that's the only way you learn.
 
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  • #2
The pieces you did look correct, but there is a danger of losing track of some pieces. It would be better to keep everything together at each step like this:
##\frac {2k^2k^3}{k^{-1}k^{-5}}(5k^{-2})^{-3}##
## = \frac {4k^5}{k^{-1}k^{-5}}(5k^{-2})^{-3}##
## = \frac {4k^5}{k^{-6}}(5k^{-2})^{-3}##
## = 4k^{11}(5k^{-2})^{-3}##
## = 4k^{11}(5^{-3}k^6)##

At this point, I don't see why you are asking your question. You use the negative power for the 5 because you have a negative, ##-3##, and you use the positive power for the ##k^{17}## because you got positive exponents: ##k^{11+6} = k^{17}##
 
  • #3
Sorry if I wasn't clear: why is it ## \frac {4k^{17}} {125}## instead of ##\frac {4} {125k^{17}}##
 
  • #4
hackedagainanda said:
Sorry if I wasn't clear: why is it ## \frac {4k^{17}} {125}## instead of ##\frac {4} {125k^{17}}##
You successfully got it to ##4k^{11}5^{-3}k^6##.
What did you do next?
 
  • #5
hackedagainanda said:
Homework Statement:: Simplify and remove all negative exponents from this equation:

## \frac {2k^2 k^3} {k^{-1} k^{-5}}(5k^{-2})^{-3}##
Relevant Equations:: Quotient Rule: ##\frac {x^a} {x^b} = x^{a-b}##

Product Rule: ## x^a x^b = x^{a+b}##

So my attempt is this: ##(2k)^2 k^3 = 4k^5## to clear the top numerator
Your attempt is not correct. From the original expression, the numerator is ##2k^2k^3##, which is ##2 \cdot k^2 \cdot k^3 = 2k^5##. In your attempt, you have ##(2k)^2k^3##. This is different from the original expression.

BTW, what you're given is not an equation -- an equation has a = symbol in it.
hackedagainanda said:
then to clear the denominator ## k^{-1} k^{-5} = k^{-6}##Then I apply the quotient rule and get ##4k^{11} (5k^{-2})^{-3}## and simplifying the right hand side I get ##5^{-3} k^6## here is where I got lost, why is it that when you use a negative exponent on the 5 to get 1/125 why do you apply the ##k^{17}## to numerator and not the denominator ##\frac {4k^{17}} {125}## Why isn't the answer ## \frac {4} {125k^{17}}## ?

I probably forgot the rules from arithmetic most likely, I feel embarrassed to ask but that's the only way you learn.
If the expression in the problem statement is the correct one (i.e., with ##2k^2## rather than ##(2k)^2 )##, then the result I get is this:
$$\frac 2 {125}k^{17}$$
 
Last edited:
  • #6
haruspex said:
You successfully got it to ##4k^{11}5^{-3}k^6##.
What did you do next?
I see now, since the first k variable is attached to the 4 I group the second instance of the k variable to the first and get ##4k^{17} {1/125}##
 
  • #7
Mark44 said:
Your attempt is not correct. From the original expression, the numerator is ##2k^2k^3##, which is ##2 \cdot k^2 \cdot k^3 = 2k^5##. In your attempt, you have ##(2k)^2k^3##. This is different from the original expression.

BTW, what you're given is not an equation -- an equation has a = symbol in it.

If the expression in the problem statement is the correct one (i.e., with ##2k^2## rather than ##(2k)^2 )##, then the result I get is this:
$$\frac 2 {125}k^{17}$$
Sorry I forgot to include the parentheses in the first expression, and you are right the problem is an expression.
 

Related to Exponents clarification question

1. What is an exponent?

An exponent is a number that represents how many times a base number is multiplied by itself. It is written as a superscript to the right of the base number, such as 23 where 2 is the base and 3 is the exponent.

2. What does an exponent clarify?

An exponent clarifies the number of times a base number is multiplied by itself. It can also clarify the magnitude of a number, as exponents can represent very large or very small numbers more efficiently than writing out all the digits.

3. How do you read an exponent?

An exponent is read as "base to the power of the exponent." For example, 23 is read as "2 to the power of 3" or "2 cubed."

4. What is the difference between an exponent and a power?

An exponent is the number that represents how many times a base number is multiplied by itself, while a power is the result of raising a base number to an exponent. For example, in 23, 3 is the exponent and 8 is the power.

5. How do you solve exponential equations?

To solve exponential equations, you can use the properties of exponents, such as the product rule, quotient rule, and power rule. You can also use logarithms to solve for the unknown variable. It is important to follow the order of operations and simplify the equation before solving for the variable.

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