The powers of 9

Algebra Level 1

9 ( 9 9 − 9 8 ) + 9 9 = 9 a , a = ? \Large 9(9^9 - 9^8) + 9^9 = 9^a \quad,\quad a=\ ?


The answer is 10.

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Jean-Luc LeBlanc by Jean-Luc LeBlanc , 28, Canada

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11 solutions

Rohit Udaiwal
Sep 22, 2015

9 10 − 9 9 + 9 9 = 9 a 9^{10}-9^9+9^9=9^a ⇒ 9 10 = 9 a ⇒ a = 10 \Rightarrow {9^{10}=9^{a}}\Rightarrow {\boxed {a=10}}

10 Helpful 1 Interesting 2 Brilliant 0 Confused

Granted that this is algebra, but logic can give a far more simplistic, yet still utterly correct, answer.

You did not specify that the value of a had to be numeric, therefore, the value of a is a . After all, if 0=0, 1=1, 2=2, etc, then a = a

Eddie Spagnoli - 5 years, 8 months ago

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[ (9^0+8^0)/2 ] = a + 1

Find the numeric value of a to find, on a scale of 1-10, how much I care. :)

Jean-Luc LeBlanc - 5 years, 8 months ago

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Clever, my friend, truly clever. ;) I would change it to "Find the numeric value of a " to stop people like me.

Eddie Spagnoli - 5 years, 8 months ago

Any number raised to power 0 is 1. So, ( 1+1)/2 = a + 1.. therefore, a = 0

Mon Grapa - 5 years, 8 months ago
Chew-Seong Cheong
Sep 24, 2015

9 ( 9 9 − 9 8 ) + 9 9 = 9 9 ( 9 − 1 ) + 9 9 = 9 9 ( 8 + 1 ) = 9 10 ⇒ a = 10 \begin{aligned} 9(9^9-9^8) + 9^9 & = 9^9(9-1)+9^9 \\ & = 9^9(8+1) \\ & = 9^{10} \\ \Rightarrow a & = \boxed{10} \end{aligned}

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Atika Samiha
Sep 22, 2015

9^10-9^9+9^9=9^a or,a=10

2 Helpful 0 Interesting 0 Brilliant 0 Confused

9 ( 9 9 − 9 8 ) + 9 9 = 9 a 9(9^9-9^8)+9^9=9^a 9 10 − 9 9 + 9 9 = 9 a 9^{10}-9^9+9^9=9^a 9 10 = 9 a 9^{10}=9^a a = 10 a= \boxed{10}

1 Helpful 1 Interesting 0 Brilliant 0 Confused
Shivangi Gupta
Oct 20, 2015

9(9^9-9^8)+9^9=9^a =9^10-9^9+9^9=9^10=9^a =a=10

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Sirajudheen Mp
Oct 19, 2015

9(9^9-9^8)+9=9^9(9-1)+9^9=(9^9)(8+1)=(9^9)9=9^(9+1)=9^10 thus a=10

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Crystal Isabel
Oct 16, 2015

9((9^9)-(9^8))+9^9= 9(9^1)+9^9= 9x9+9^9= 9^11

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Ervin Cunanan
Oct 2, 2015

Basically just look at the exponent to left to right And think of this equation 9-8+9=10

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Khalid Basheer
Sep 25, 2015

9*9^8(9^1-1)+9^9=9^a... as bases are same then we can add the powers 9^9(8)+9^9=9^a 9^9(8+1)=9^a.. 9^9(9)=9^a... 9^10=9^a. as bases are equal so their exponents are also equal. a=10 is answer

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Hadia Qadir
Sep 23, 2015

9(9^9 - 9^8) + 9^9 = 9^a

9^10 - 9^9 + 9^9 = 9^a

9^10 = 9^a

10=a

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Hamza Muhammad
Sep 22, 2015

9 x ( 9^9 - 9^8 ) + 9^9 = 9^9 (9-1) + 9^9 = 9^9( 9 - 1 + 1 ) = 9^9 ( 9 ) = 9^10 :D

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