Mixing
Let $a = frac{9+sqrt{45}}{2}$. Find $frac{1}{a}$
$begingroup$
I've been wrapping my head around this question lately:
Let
$$a = frac{9+sqrt{45}}{2}$$
Find the value of
$$frac{1}{a}$$
I've done it like this:
$$frac{1}{a}=frac{2}{9+sqrt{45}}$$
I rationalize the denominator like this:
$$frac{1}{a}=frac{2}{9+sqrt{45}} times (frac{9-sqrt{45}}{9-sqrt{45}})$$
This is what I should get:
$$frac{1}{a} = frac{2(9-sqrt{45})}{81-45} rightarrow frac{1}{a}=frac{18-2sqrt{45}}{36})$$
Which I can simplify to:
$$frac{1}{a}=frac{2(9-sqrt{45})}{36}rightarrowfrac{1}{a}=frac{9-sqrt{45}}{18}$$
However, this answer can't be found in my multiple choice question here:
Any hints on what I'm doing wrong?
fractions
asked Aug 27 '17 at 17:01
CesareCesare
777410
$endgroup$
add a comment |
$begingroup$
I've been wrapping my head around this question lately:
Let
$$a = frac{9+sqrt{45}}{2}$$
Find the value of
$$frac{1}{a}$$
I've done it like this:
$$frac{1}{a}=frac{2}{9+sqrt{45}}$$
I rationalize the denominator like this:
$$frac{1}{a}=frac{2}{9+sqrt{45}} times (frac{9-sqrt{45}}{9-sqrt{45}})$$
This is what I should get:
$$frac{1}{a} = frac{2(9-sqrt{45})}{81-45} rightarrow frac{1}{a}=frac{18-2sqrt{45}}{36})$$
Which I can simplify to:
$$frac{1}{a}=frac{2(9-sqrt{45})}{36}rightarrowfrac{1}{a}=frac{9-sqrt{45}}{18}$$
However, this answer can't be found in my multiple choice question here:
Any hints on what I'm doing wrong?
fractions
asked Aug 27 '17 at 17:01
CesareCesare
777410
$endgroup$
1
$begingroup$
Neat @Peter, thanks!
$endgroup$
– Cesare
Aug 27 '17 at 17:06
$begingroup$
Your result can easily be transformed into the expression in $b$. Just divide numerator and denominator by $3$
$endgroup$
– Peter
Aug 27 '17 at 17:07
$begingroup$
If you want to, you can copy and paste your 1st comment as an answer to my question and I'll be happy to +1 to thank you for your help.
$endgroup$
– Cesare
Aug 27 '17 at 17:09
1
$begingroup$
Why on earth did someone downvote this question ?
$endgroup$
– Peter
Aug 27 '17 at 17:17
add a comment |
$begingroup$
I've been wrapping my head around this question lately:
Let
$$a = frac{9+sqrt{45}}{2}$$
Find the value of
$$frac{1}{a}$$
I've done it like this:
$$frac{1}{a}=frac{2}{9+sqrt{45}}$$
I rationalize the denominator like this:
$$frac{1}{a}=frac{2}{9+sqrt{45}} times (frac{9-sqrt{45}}{9-sqrt{45}})$$
This is what I should get:
$$frac{1}{a} = frac{2(9-sqrt{45})}{81-45} rightarrow frac{1}{a}=frac{18-2sqrt{45}}{36})$$
Which I can simplify to:
$$frac{1}{a}=frac{2(9-sqrt{45})}{36}rightarrowfrac{1}{a}=frac{9-sqrt{45}}{18}$$
However, this answer can't be found in my multiple choice question here:
Any hints on what I'm doing wrong?
fractions
asked Aug 27 '17 at 17:01
CesareCesare
777410
$endgroup$
I've been wrapping my head around this question lately:
Let
$$a = frac{9+sqrt{45}}{2}$$
Find the value of
$$frac{1}{a}$$
I've done it like this:
$$frac{1}{a}=frac{2}{9+sqrt{45}}$$
I rationalize the denominator like this:
$$frac{1}{a}=frac{2}{9+sqrt{45}} times (frac{9-sqrt{45}}{9-sqrt{45}})$$
This is what I should get:
$$frac{1}{a} = frac{2(9-sqrt{45})}{81-45} rightarrow frac{1}{a}=frac{18-2sqrt{45}}{36})$$
Which I can simplify to:
$$frac{1}{a}=frac{2(9-sqrt{45})}{36}rightarrowfrac{1}{a}=frac{9-sqrt{45}}{18}$$
However, this answer can't be found in my multiple choice question here:
Any hints on what I'm doing wrong?
fractions
fractions
asked Aug 27 '17 at 17:01
CesareCesare
777410
asked Aug 27 '17 at 17:01
CesareCesare
777410
asked Aug 27 '17 at 17:01
CesareCesare
777410
asked Aug 27 '17 at 17:01
CesareCesare
777410
asked Aug 27 '17 at 17:01
CesareCesare
777410
777410
1
$begingroup$
Neat @Peter, thanks!
$endgroup$
– Cesare
Aug 27 '17 at 17:06
$begingroup$
Your result can easily be transformed into the expression in $b$. Just divide numerator and denominator by $3$
$endgroup$
– Peter
Aug 27 '17 at 17:07
$begingroup$
If you want to, you can copy and paste your 1st comment as an answer to my question and I'll be happy to +1 to thank you for your help.
$endgroup$
– Cesare
Aug 27 '17 at 17:09
1
$begingroup$
Why on earth did someone downvote this question ?
$endgroup$
– Peter
Aug 27 '17 at 17:17
add a comment |
1
$begingroup$
Neat @Peter, thanks!
$endgroup$
– Cesare
Aug 27 '17 at 17:06
$begingroup$
Your result can easily be transformed into the expression in $b$. Just divide numerator and denominator by $3$
$endgroup$
– Peter
Aug 27 '17 at 17:07
$begingroup$
If you want to, you can copy and paste your 1st comment as an answer to my question and I'll be happy to +1 to thank you for your help.
$endgroup$
– Cesare
Aug 27 '17 at 17:09
1
$begingroup$
Why on earth did someone downvote this question ?
$endgroup$
– Peter
Aug 27 '17 at 17:17
1
1
$begingroup$
Neat @Peter, thanks!
$endgroup$
– Cesare
Aug 27 '17 at 17:06
$begingroup$
Neat @Peter, thanks!
$endgroup$
– Cesare
Aug 27 '17 at 17:06
$begingroup$
Your result can easily be transformed into the expression in $b$. Just divide numerator and denominator by $3$
$endgroup$
– Peter
Aug 27 '17 at 17:07
$begingroup$
Your result can easily be transformed into the expression in $b$. Just divide numerator and denominator by $3$
$endgroup$
– Peter
Aug 27 '17 at 17:07
$begingroup$
If you want to, you can copy and paste your 1st comment as an answer to my question and I'll be happy to +1 to thank you for your help.
$endgroup$
– Cesare
Aug 27 '17 at 17:09
$begingroup$
If you want to, you can copy and paste your 1st comment as an answer to my question and I'll be happy to +1 to thank you for your help.
$endgroup$
– Cesare
Aug 27 '17 at 17:09
1
1
$begingroup$
Why on earth did someone downvote this question ?
$endgroup$
– Peter
Aug 27 '17 at 17:17
$begingroup$
Why on earth did someone downvote this question ?
$endgroup$
– Peter
Aug 27 '17 at 17:17
add a comment |
4 Answers
4
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oldest
votes
$begingroup$
$$frac { 1 }{ a } =frac { 9-sqrt { 45 } }{ 18 } =frac { 3left( 3-sqrt { 5 } right) }{ 18 } =frac { 3-sqrt { 5 } }{ 6 } $$
answered Aug 27 '17 at 17:04
haqnaturalhaqnatural
20.8k72457
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add a comment |
$begingroup$
Since it is a multiple choice question, you can just multiply the given term with all the possible terms and verify when you get $1$
answered Aug 27 '17 at 17:14
PeterPeter
48.3k1139134
$endgroup$
add a comment |
$begingroup$
The answer is b, which is equivalent to your answer after simplifying.
This is because $ 9 - sqrt{45} = 9 - sqrt{9 * 5} = 9 - 3sqrt{5}$. Then,$ frac{9 - 3sqrt{5}}{18} = frac{3 - sqrt{5}}{6}. $
answered Aug 27 '17 at 17:06
Vijay RamanujanVijay Ramanujan
456
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add a comment |
$begingroup$
Since the $ 45 $ inside the radical is expressed a product containing a perfect square ($ 9 $), $$ sqrt{45} = sqrt{9 times 5} = sqrt{9}sqrt{5} = 3sqrt{5} $$
Thus, $$ frac{1}{a} = frac{9 - sqrt{45}}{18} = frac{9 - 3sqrt{5}}{18} $$
Dividing the numerator and denominator by $ 3 $ (the greatest common factor of the coefficients $ 3 $ and $ 9 $ in the numerator and $ 18 $ in the denominator):
$$ frac{(9 - 3sqrt{5}) / 3}{18 / 3} = frac{3 - sqrt{5}}{6} $$ which is one of the choices (choice (b)).
answered Jan 20 at 21:14
Marvin CohenMarvin Cohen
158117
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add a comment |
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4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
$$frac { 1 }{ a } =frac { 9-sqrt { 45 } }{ 18 } =frac { 3left( 3-sqrt { 5 } right) }{ 18 } =frac { 3-sqrt { 5 } }{ 6 } $$
answered Aug 27 '17 at 17:04
haqnaturalhaqnatural
20.8k72457
$endgroup$
add a comment |
$begingroup$
$$frac { 1 }{ a } =frac { 9-sqrt { 45 } }{ 18 } =frac { 3left( 3-sqrt { 5 } right) }{ 18 } =frac { 3-sqrt { 5 } }{ 6 } $$
answered Aug 27 '17 at 17:04
haqnaturalhaqnatural
20.8k72457
$endgroup$
add a comment |
$begingroup$
$$frac { 1 }{ a } =frac { 9-sqrt { 45 } }{ 18 } =frac { 3left( 3-sqrt { 5 } right) }{ 18 } =frac { 3-sqrt { 5 } }{ 6 } $$
answered Aug 27 '17 at 17:04
haqnaturalhaqnatural
20.8k72457
$endgroup$
$$frac { 1 }{ a } =frac { 9-sqrt { 45 } }{ 18 } =frac { 3left( 3-sqrt { 5 } right) }{ 18 } =frac { 3-sqrt { 5 } }{ 6 } $$
answered Aug 27 '17 at 17:04
haqnaturalhaqnatural
20.8k72457
answered Aug 27 '17 at 17:04
haqnaturalhaqnatural
20.8k72457
answered Aug 27 '17 at 17:04
haqnaturalhaqnatural
20.8k72457
answered Aug 27 '17 at 17:04
haqnaturalhaqnatural
20.8k72457
20.8k72457
add a comment |
add a comment |
$begingroup$
Since it is a multiple choice question, you can just multiply the given term with all the possible terms and verify when you get $1$
answered Aug 27 '17 at 17:14
PeterPeter
48.3k1139134
$endgroup$
add a comment |
$begingroup$
Since it is a multiple choice question, you can just multiply the given term with all the possible terms and verify when you get $1$
answered Aug 27 '17 at 17:14
PeterPeter
48.3k1139134
$endgroup$
add a comment |
$begingroup$
Since it is a multiple choice question, you can just multiply the given term with all the possible terms and verify when you get $1$
answered Aug 27 '17 at 17:14
PeterPeter
48.3k1139134
$endgroup$
Since it is a multiple choice question, you can just multiply the given term with all the possible terms and verify when you get $1$
answered Aug 27 '17 at 17:14
PeterPeter
48.3k1139134
answered Aug 27 '17 at 17:14
PeterPeter
48.3k1139134
answered Aug 27 '17 at 17:14
PeterPeter
48.3k1139134
answered Aug 27 '17 at 17:14
PeterPeter
48.3k1139134
48.3k1139134
add a comment |
add a comment |
$begingroup$
The answer is b, which is equivalent to your answer after simplifying.
This is because $ 9 - sqrt{45} = 9 - sqrt{9 * 5} = 9 - 3sqrt{5}$. Then,$ frac{9 - 3sqrt{5}}{18} = frac{3 - sqrt{5}}{6}. $
answered Aug 27 '17 at 17:06
Vijay RamanujanVijay Ramanujan
456
$endgroup$
add a comment |
$begingroup$
The answer is b, which is equivalent to your answer after simplifying.
This is because $ 9 - sqrt{45} = 9 - sqrt{9 * 5} = 9 - 3sqrt{5}$. Then,$ frac{9 - 3sqrt{5}}{18} = frac{3 - sqrt{5}}{6}. $
answered Aug 27 '17 at 17:06
Vijay RamanujanVijay Ramanujan
456
$endgroup$
add a comment |
$begingroup$
The answer is b, which is equivalent to your answer after simplifying.
This is because $ 9 - sqrt{45} = 9 - sqrt{9 * 5} = 9 - 3sqrt{5}$. Then,$ frac{9 - 3sqrt{5}}{18} = frac{3 - sqrt{5}}{6}. $
answered Aug 27 '17 at 17:06
Vijay RamanujanVijay Ramanujan
456
$endgroup$
The answer is b, which is equivalent to your answer after simplifying.
This is because $ 9 - sqrt{45} = 9 - sqrt{9 * 5} = 9 - 3sqrt{5}$. Then,$ frac{9 - 3sqrt{5}}{18} = frac{3 - sqrt{5}}{6}. $
answered Aug 27 '17 at 17:06
Vijay RamanujanVijay Ramanujan
456
answered Aug 27 '17 at 17:06
Vijay RamanujanVijay Ramanujan
456
answered Aug 27 '17 at 17:06
Vijay RamanujanVijay Ramanujan
456
answered Aug 27 '17 at 17:06
Vijay RamanujanVijay Ramanujan
456
456
add a comment |
add a comment |
$begingroup$
Since the $ 45 $ inside the radical is expressed a product containing a perfect square ($ 9 $), $$ sqrt{45} = sqrt{9 times 5} = sqrt{9}sqrt{5} = 3sqrt{5} $$
Thus, $$ frac{1}{a} = frac{9 - sqrt{45}}{18} = frac{9 - 3sqrt{5}}{18} $$
Dividing the numerator and denominator by $ 3 $ (the greatest common factor of the coefficients $ 3 $ and $ 9 $ in the numerator and $ 18 $ in the denominator):
$$ frac{(9 - 3sqrt{5}) / 3}{18 / 3} = frac{3 - sqrt{5}}{6} $$ which is one of the choices (choice (b)).
answered Jan 20 at 21:14
Marvin CohenMarvin Cohen
158117
$endgroup$
add a comment |
$begingroup$
Since the $ 45 $ inside the radical is expressed a product containing a perfect square ($ 9 $), $$ sqrt{45} = sqrt{9 times 5} = sqrt{9}sqrt{5} = 3sqrt{5} $$
Thus, $$ frac{1}{a} = frac{9 - sqrt{45}}{18} = frac{9 - 3sqrt{5}}{18} $$
Dividing the numerator and denominator by $ 3 $ (the greatest common factor of the coefficients $ 3 $ and $ 9 $ in the numerator and $ 18 $ in the denominator):
$$ frac{(9 - 3sqrt{5}) / 3}{18 / 3} = frac{3 - sqrt{5}}{6} $$ which is one of the choices (choice (b)).
answered Jan 20 at 21:14
Marvin CohenMarvin Cohen
158117
$endgroup$
add a comment |
$begingroup$
Since the $ 45 $ inside the radical is expressed a product containing a perfect square ($ 9 $), $$ sqrt{45} = sqrt{9 times 5} = sqrt{9}sqrt{5} = 3sqrt{5} $$
Thus, $$ frac{1}{a} = frac{9 - sqrt{45}}{18} = frac{9 - 3sqrt{5}}{18} $$
Dividing the numerator and denominator by $ 3 $ (the greatest common factor of the coefficients $ 3 $ and $ 9 $ in the numerator and $ 18 $ in the denominator):
$$ frac{(9 - 3sqrt{5}) / 3}{18 / 3} = frac{3 - sqrt{5}}{6} $$ which is one of the choices (choice (b)).
answered Jan 20 at 21:14
Marvin CohenMarvin Cohen
158117
$endgroup$
Since the $ 45 $ inside the radical is expressed a product containing a perfect square ($ 9 $), $$ sqrt{45} = sqrt{9 times 5} = sqrt{9}sqrt{5} = 3sqrt{5} $$
Thus, $$ frac{1}{a} = frac{9 - sqrt{45}}{18} = frac{9 - 3sqrt{5}}{18} $$
Dividing the numerator and denominator by $ 3 $ (the greatest common factor of the coefficients $ 3 $ and $ 9 $ in the numerator and $ 18 $ in the denominator):
$$ frac{(9 - 3sqrt{5}) / 3}{18 / 3} = frac{3 - sqrt{5}}{6} $$ which is one of the choices (choice (b)).
answered Jan 20 at 21:14
Marvin CohenMarvin Cohen
158117
answered Jan 20 at 21:14
Marvin CohenMarvin Cohen
158117
answered Jan 20 at 21:14
Marvin CohenMarvin Cohen
158117
answered Jan 20 at 21:14
Marvin CohenMarvin Cohen
158117
158117
add a comment |
add a comment |
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1
$begingroup$
Neat @Peter, thanks!
$endgroup$
– Cesare
Aug 27 '17 at 17:06
$begingroup$
Your result can easily be transformed into the expression in $b$. Just divide numerator and denominator by $3$
$endgroup$
– Peter
Aug 27 '17 at 17:07
$begingroup$
If you want to, you can copy and paste your 1st comment as an answer to my question and I'll be happy to +1 to thank you for your help.
$endgroup$
– Cesare
Aug 27 '17 at 17:09
1
$begingroup$
Why on earth did someone downvote this question ?
$endgroup$
– Peter
Aug 27 '17 at 17:17