chrisGA
  • chrisGA
Evaluate fourth root of 9 multiplied by square root of 9 over the fourth root of 9 to the power of 5.
Mathematics
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SOLVED
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chestercat
  • chestercat
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chrisGA
  • chrisGA
9 to the power of negative 1 over 2 9 to the power of negative 1 over 4 9 92
anonymous
  • anonymous
If I understand correctly, the question is to simplify\[\frac{ 9^{\frac{ 1 }{ 4 }} \times 9^{\frac{ 1 }{ 2 }} }{ \left( 9^{\frac{ 1 }{ 4 }} \right)^{5} }\]Is this correct?
chrisGA
  • chrisGA
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anonymous
  • anonymous
OK. When working with exponents, many times it is easier to express them as fractions rather than radicals. My statement is equivalent to yours. Do you understand the fractional exponents?
chrisGA
  • chrisGA
yes a lil
anonymous
  • anonymous
For example,\[\sqrt{x}=x ^{\frac{ 1 }{ 2 }}\]\[\sqrt[3]{x}=x ^{\frac{ 1 }{ 3 }}\]\[\sqrt[4]{x}=x ^{\frac{ 1 }{ 4 }}\]\[\sqrt[4]{x ^{5}}=\left( x ^{5} \right)^{\frac{ 1 }{ 4 }}=x ^{\frac{ 5 }{ 4 }}\]
anonymous
  • anonymous
So, with the expression written with fractional exponents, simplify the numerator first. The rule when multiplying powers with the same base is to ADD the exponents. Then, divide the numerator by the denominator, The rule when dividing powers with the same base is that you SUBTRACT the exponents. This help?
chrisGA
  • chrisGA
it does a lil bit what would the answer be?
anonymous
  • anonymous
For example,\[\frac{ 3^{\frac{ 1 }{ 4 }}\times 3^{\frac{ 1 }{ 2 }} }{ 3^{\frac{ 7 }{ 4 }} }=\frac{ 3^{\frac{ 3 }{ 4 }} }{ 3^{\frac{ 7 }{ 4 }} }=3^\left( \frac{ 3 }{ 4 } -\frac{ 7 }{ 4 }\right)=3^{-\frac{ 4 }{ 4 }}=3^{-1}\]
anonymous
  • anonymous
Sorry for taking so long with the typing. I can't give you the answer, but I can check yours. What do you get?
chrisGA
  • chrisGA
it ok and i got B? is that correct
anonymous
  • anonymous
Don't think so. Let's take it one step at a time. What do you get when you simplify just the numerator?
chrisGA
  • chrisGA
9
anonymous
  • anonymous
\[9^{\frac{ 1 }{ 4 }}\times 9^{\frac{ 1 }{ 2 }} = 9^{\left( \frac{ 1 }{ 4 } +\frac{ 1 }{ 2 }\right)}=?\]
chrisGA
  • chrisGA
is it C?
anonymous
  • anonymous
Don't think so. How about simplifying the numerator first, as above. What would you get?
anonymous
  • anonymous
What is 1/4 + 1/2 ?
chrisGA
  • chrisGA
3/4
anonymous
  • anonymous
Hello? Do you still want my help?
chrisGA
  • chrisGA
3/4
anonymous
  • anonymous
Right on! So the numerator simplifies to\[9^{\frac{ 3 }{ 4 }}\]Now simplify the denominator. It is a power raise to another exponent. The rule in this case is to MULTIPLY the exponents. So\[\left(9^{\frac{ 1 }{ 4 }} \right)^{5}=9^{?}\] What do you get?
chrisGA
  • chrisGA
2?
anonymous
  • anonymous
No. Keep the base (9) and multiply just the exponents together.\[\left( 9^{\frac{ 1 }{ 4 }} \right)^{5} = 9^{\left( \frac{ 1 }{ 4 } \times5\right)} = 9^{?}\]
anonymous
  • anonymous
In other words, what is 1/4 x 5 ?
chrisGA
  • chrisGA
5/4?
anonymous
  • anonymous
Perfect. Now your problem has been simplified to \[\frac{ 9^{\frac{ 3 }{ 4 }} }{ 9^{\frac{ 5 }{ 4 }} }\]When dividing powers with the same base, the rule is to keep the base and SUBTRACT the exponents. Therefore\[\frac{ 9^{\frac{ 3 }{ 4 }} }{ 9^{\frac{ 5 }{ 4 }} } = 9^{\left( \frac{ 3 }{ 4 } -\frac{ 5 }{ 4 }\right)} = 9^{?}\]
anonymous
  • anonymous
In other words, what is 3/4 - 5/4 ?
chrisGA
  • chrisGA
-1/2
anonymous
  • anonymous
Terrific. That's your answer.\[9^{-\frac{ 1 }{ 2 }}\] Lot of small step. And lots of rules for working with exponents. Well done!
chrisGA
  • chrisGA
thank you very much
anonymous
  • anonymous
You're welcome. A bit of practice, and you'll be a master.

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