2002 Math OSK, Number 1: Transposing Evens

Algebra Level 1

( 2 4 ) 8 ÷ ( 4 8 ) 2 = ? \LARGE (\color{#D61F06}2^{\color{#20A900}4})^{\color{#3D99F6}8} \div ({\color{#20A900}4}^{\color{#3D99F6}8})^{\color{#D61F06}2} = \ \color{grey}?

1 4 \frac { 1 }{ 4 } 1 1 8 8 2 2 1 2 \frac { 1 }{ 2 }

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Abyoso Hapsoro by Abyoso Hapsoro , 22, Indonesia

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31 solutions

Caleb Townsend
Apr 4, 2015

( 2 4 ) 8 ÷ ( 4 8 ) 2 = 2 32 ÷ ( 4 8 ) 2 = 2 32 ÷ ( ( 2 2 ) 8 ) 2 = 2 32 ÷ ( 2 16 ) 2 = 2 32 ÷ 2 32 = 1 (2^4)^8\div (4^8)^2 = 2^{32}\div (4^8)^2 \\ = 2^{32}\div ((2^2)^8)^2 \\ = 2^{32}\div (2^{16})^2 \\ = 2^{32}\div 2^{32} \\ = 1

96 Helpful 3 Interesting 3 Brilliant 6 Confused

(2^4)^8÷(4^8)^2= 2^32÷4^16= 1/2^16 what are you people doing? Two exponents next to each other such as (x^4)^8, you multiply the exponents. And then when dividing you subtract the eponent in the denominator from the exponent in the numerator. In this case 32-16, and then you divide your problem and the answer is 1/2^16

Tara Newman - 5 years, 4 months ago

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You have to reduce the 4 to 2^2 first, then it becomes 32-32

k RNA - 5 years, 3 months ago

thx.... v much

Tim Kenny - 6 years, 2 months ago

ya i forget it a little bit but am serprizaed to see answer

Aadil Naeem - 6 years, 1 month ago

2^4 = 16 16^8 = 4.2949673e+009 4^8 = 65536 sqr(65536) = 4.2949673e+009 4.2949673e+009/4.2949673e+009 = 1

Muthu Cp - 6 years, 1 month ago

2^4 = 16 16^8 = 4.2949673+009 4^8 = 65536 (65536) = 4.2949673+009 4.2949673+009/4.2949673+009 = 1

Muthu Cp - 6 years, 1 month ago
Ubaidullah Khan
Apr 8, 2015

Both gets cancelled when multiplies using laws of indices. Hence, we get 1 as our answer.

6 Helpful 0 Interesting 0 Brilliant 0 Confused
Achille 'Gilles'
Dec 14, 2015

3 Helpful 1 Interesting 2 Brilliant 0 Confused
Lok Diesel
Apr 8, 2015

((2^4)^8)÷((4^2)^8) ((2^4)^8)÷((2^(2+2))^8) ((2^4)^8)÷((2^4)^8) =1

3 Helpful 0 Interesting 0 Brilliant 0 Confused
Manan Agarwal
Apr 11, 2015

(2^4)^8=2^32 (4^8)^2=4^16=2^2^16=2^32

2^32-2^32 =2^1 =0

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Felipe Faria
Apr 18, 2015

Just to add another solution to the pile. You are able to solve this problem utilizing log ⁡ 2 \log_2 and the basic understanding of exponents.

( 2 4 ) 8 = 2 8 ∗ 4 = 2 32 (2^{4})^{8} = 2^{8*4} = 2^{32}

( 4 8 ) 2 = 4 8 ∗ 2 = 4 16 (4^{8})^{2} = 4^{8*2} = 4^{16}

log ⁡ 2 2 32 l o g 2 4 16 \frac{\log_2 2^{32}}{log_2 4^{16}}

32 log ⁡ 2 2 16 log ⁡ 2 4 \frac{32\log_2 2}{16\log_2 4}

( 32 ∗ 1 ) ( 16 ∗ 2 ) \frac{(32*1)}{(16 * 2)}

32 32 = 1 \frac{32}{32} = 1

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Just observe that 4=2^2, so (2^4)^8=((2^2)^2)^8

1 Helpful 0 Interesting 0 Brilliant 0 Confused

work inside the parenthesis first . . . . . .

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Ram Padmanabhan
Jun 10, 2015

(2^4)^8 \divides (4^8)^2 =(2^4)^8 \divides (4^2)^8 =1 (Because 2^4 is the same as 4^2)

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Sarvocch Gupta
May 24, 2015

(2^4)^8÷ (4^8)^2

=2^32 ÷ 4^16

=(2^2)^16 ÷ 4^16

=4^16 ÷ 4^16

=1

0 Helpful 0 Interesting 0 Brilliant 0 Confused

[(2^4)^8=(16)^8]/ [(4^8)^2=(4^2)^8=(16)^8]

16^8/16^8=1

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Shesha Patel
Apr 22, 2015

2^32/4^16=2^32-2^16-2^16=2^0=1

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Wael Hagag
Apr 19, 2015

All exponents are multiplied together in the nominator and the denominator

0 Helpful 0 Interesting 0 Brilliant 1 Confused
Arbu Patel
Apr 19, 2015

2^32/4^16.
=16[2]/16 [2] =1

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Vukosi Mashaba
Apr 19, 2015

Let,8 be =x,therefore:(2^4)^x ÷ (4^x)^2 ,2^4x ÷2^4x,then,just divide the two,an get 1

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Aakash Patel
Apr 19, 2015

2^32/4^16=2^32/2^32=1

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Ankith Ki
Apr 19, 2015

2^32/2^32=1

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Rubina Shaheen
Apr 18, 2015

Something important for our youngs is to reinforce that we get "1" after cancellation of all numbers WHEN we are in process of multiplication / division... but we get "0" if we do the same in addition /subtraction.

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Sai Durga Charan
Apr 18, 2015

(2^4)^8/(4^8)^2=(4^2)^8/(4^8)^2=4^16/4^16=1

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Mudassir Hassan
Apr 16, 2015

All others have done brilliantly.. But what I've done is just see the order of the nbrs as 2,4,8 divided by the same nbr but order is changed 4,8,2 so i thought if the nbrs are same they must be equal to 1. ;-)

0 Helpful 0 Interesting 0 Brilliant 0 Confused

I am afraid that dosent work in all scenarios.

Take for example.

( 2 5 ) 8 ≠ ( 8 5 ) 2 (2^{5})^{8} \neq (8^{5})^{2}

Felipe Faria - 6 years, 1 month ago
Anush Anand
Apr 12, 2015

(2^4)^8 / (4^8)^2 = 2^32 / 4^16 Simplified, 2^32 / ((2^2)^8)^2 = 2^32 / 2^32 = 1

0 Helpful 0 Interesting 0 Brilliant 0 Confused

= [ (2^4)^8 ] / [ (4^8)^2 ] = [ (4^2)^8 ] / [ (4^8)^2 ] = Law of Exponents (Power raised to a power) --> 4^16 / 4^16 = 1 :)

0 Helpful 0 Interesting 0 Brilliant 0 Confused

2^4=4^2 So,by substituting one of them, it can be easily proven.

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Moderator note:

Cute! You could have phrased your solution better like this:

Because 2 4 = 4 2 2^4 = 4^2 , then ( 2 4 ) 8 = ( 4 2 ) 8 = ( 4 8 ) 2 (2^4)^8 = (4^2)^8 = (4^8)^2 by properties of indices.

Mark Johnson
Apr 11, 2015

(2 * 2 * 2 * 2)^8 ÷ (4)^16 = ((4)^2)^8) ÷ (4)^16 = (4)^16 ÷ (4)^16 = 4^0 =1 OR (2)^4 ÷ (4)^2 = 16 ÷ 16 = 1 OR several other variations using the Laws of Exponents.

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Cindy Pelley
Apr 11, 2015

(2^4)^8 = 4294967296. (4^8)^2 = 4294967296

4294967296/4294967296 = 1

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Moderator note:

It would be impractical to calculate large indices especially when one could easily simplify the work through the properties of indices.

You don't need to calculate them, this needed calculator or long time counting a needless number.

Abyoso Hapsoro - 6 years, 2 months ago

(2^4 )^8÷(4^8 )^2=(4^2 )^8÷(4^8 )^2=4^16÷4^16=4^(16-16)=4^0=1

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Roger Millo
Apr 11, 2015

2^4+8 ÷2^2+8+2= 2^12÷2^12= 1

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Jeff Capaldo
Apr 11, 2015

Solve for the powers within parentheses. After that, you're all set. Both sides of the equation are equal. Divided, the answer is 1.

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Ana Vallado
Apr 11, 2015

Cancel both 8 exponents and you'll get 2^4 / 4^2 which is 16/16=1

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Cam Cope
Apr 11, 2015

Honestly, I just canceled the ^8 s out and solved from 16/16 which equals 1

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Sayed M Amir
Apr 11, 2015

(2^4)^8 ÷ (4^8)^2 (2^4)^8 ÷ [4^2]^8 (2^4)^8 ÷ [(2×2)^2]^8 (2^4)^8 ÷ [(2^2)^2]^8 (2^4)^8 ÷ [2^2×2]^8 { BY LAW OF INDICES} (2^4)^8 ÷ [2^4]^8 =1

0 Helpful 0 Interesting 0 Brilliant 0 Confused

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