Dealing with exponent laws (2 simple questions in one thread)

In summary, the conversation discusses simplifying and evaluating a complex expression involving exponents, as well as comparing the sizes of two large numbers written in exponential form. The conversation also includes a helpful explanation of how to combine fractions and a reminder about the properties of exponents.
  • #1
mike_302
103
0

Homework Statement



6^1+6^-1 / 6^1-6^-1 (Question is to evaluate that, but I am going to venture to guess that we are supposed to somehow simplify the question a lot further. We just finished learning all the exponent laws)


Explain how you can tell which is bigger without evaluating: 20^100 or 400^20 ?



Have not been able to evaluate at all.
 
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  • #2
x^-1 = 1/x

From there, you can finish it a number of ways. I'd probably just combine the fraction & then divide the whole thing to get a fraction result.

To find which is larger, they both start with 6, & either add or subtract x^-1. If you add a positive number to 6 or subtract a negative number, the positive number will be larger.
 
  • #3
OH! haha. The second part with "which is bigger" is a whole differnt question. Sorry I didn't make that clear :P .

Anyways, since only the fist question was answered with accuracy here, I would like to discus that quickly. How do you combine the fraction like you say? That is where I am getting mixed up: Rearranging to get all positive exponents.
 
  • #4
mike_302 said:

Homework Statement



6^1+6^-1 / 6^1-6^-1 (Question is to evaluate that, but I am going to venture to guess that we are supposed to somehow simplify the question a lot further. We just finished learning all the exponent laws)
[tex]\frac{6+ 6^{-1}}{6- 6^{-1}}[/tex]
Multiply both numerator and denominator by 6.


Explain how you can tell which is bigger without evaluating: 20^100 or 400^20 ?
[itex]20= 2^2(5)[/itex] so [itex]20^{100}= 2^{200}(5^{100})[/itex]. [itex]400= 40(100)= (8*5)(4*25)= 2^5(5^3)[/itex] so [itex]400^{20}= 2^{100}(5^{60})[/itex]
Can you compare those?



Have not been able to evaluate at all.
 
  • #5
ahhh! Yes, I see for both now. I understand the first one and well... The second one, I get the idea of making them both have similar bases but how you did it would take a little more concentration on my behalf. I will do that after posting this but I must thank you for your work.
 
  • #6
HallsofIvy said:
[itex]20= 2^2(5)[/itex] so [itex]20^{100}= 2^{200}(5^{100})[/itex]. [itex]400= 40(100)= (8*5)(4*25)= 2^5(5^3)[/itex] so [itex]400^{20}= 2^{100}(5^{60})[/itex]
Can you compare those?

Actually, it is easier to solve if you see that [itex]400 = 20^2[/itex]. from which you get that [itex]400^{20} = 20^{2 * 20} = 20^{40}[/itex] and of course [itex]20^{100} > 20^{40}[/itex].

EDIT: It's a property of exponents that [itex](a^b)^c = a^{bc}[/itex]
 
Last edited:
  • #7
You're right. I understand that now, no problem. Thanks for your help as well! :)
 

Related to Dealing with exponent laws (2 simple questions in one thread)

1. What are the basic exponent laws?

The basic exponent laws are the product law, quotient law, power law, and zero and negative exponent laws. These laws govern how exponents behave in mathematical expressions.

2. How do I simplify expressions with exponents?

To simplify expressions with exponents, you can use the exponent laws to combine like terms and reduce the expression to its simplest form. This involves applying the product and quotient laws to expand or condense the expression, and using the power law to simplify terms with the same base.

3. What is the difference between a positive and negative exponent?

A positive exponent indicates repeated multiplication, while a negative exponent indicates repeated division. For example, 23 means 2 multiplied by itself three times (2 x 2 x 2), while 2-3 means 2 divided by itself three times (2 ÷ 2 ÷ 2).

4. How do I handle zero exponents?

A zero exponent indicates that the base is raised to the power of 0, which always results in 1. Therefore, any term with a zero exponent can be simplified to 1.

5. Can I use exponent laws with variables?

Yes, exponent laws can be applied to expressions with variables as long as the variables have the same base. For example, x2 * x3 can be simplified to x5 using the product law.

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