Six six to the six

Algebra Level 1

6 6 + 6 6 + 6 6 + 6 6 + 6 6 + 6 6 = 6 a \large\color{#20A900} 6^{\color{#3D99F6} 6} + \color{#20A900} 6^{\color{#3D99F6} 6} + \color{#20A900} 6^{\color{#3D99F6} 6} + \color{#20A900} 6^{\color{#3D99F6} 6} + \color{#20A900} 6^{\color{#3D99F6} 6} + \color{#20A900} 6^{\color{#3D99F6} 6} = \color{#20A900}6^ {\color{#624F41}a}

If a \color{#624F41} a satisfies the equation above, what is the value of a \color{#624F41} a ?

36 36 7 7 6 6 42 42 49 49

Adrian Gonzalez by Adrian Gonzalez , 31, Mexico

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25 solutions

6 6 + 6 6 + 6 6 + 6 6 + 6 6 + 6 6 = 6 × 6 6 = 6 6 + 1 = 6 7 \Large {6}^{6}+{6}^{6}+{6}^{6}+{6}^{6}+{6}^{6}+{6}^{6}=6 \times {6}^{6} = 6^{6 + 1} = {6}^{7}

342 Helpful 20 Interesting 14 Brilliant 9 Confused

What is anyone's understanding is the

A is number of repeats And the other we copy

6^6 repeat six times ( a = 6 )

A x 6^6
6^6 . 6^1 = 6^7

This truth is an a simple example.

3² + 3² + 3² = (Obvius is 27)

But in shape used

A = 3
X = 3²

{A.x}

3² . 3 = 3³ the same 27.

This shape is perfectly right.

kauan horvath - 5 years, 2 months ago

1st we take common 6^6. 6^6[1+1+1+1+1+1]=6^a 6^6 [6]=6^a as 6 power 6 is multipling with 6 therefore we shift it on the other side and divide it 6=[6^a] / [6^6] 6= [6^a] [6^-6] base same power will be add 6^1=6^a-6 as the bases are same we take powers 1=a-6 1+6=a a=7 answer

Syed Israr Ullah Shah - 5 years, 5 months ago

Think of it as if you're multiplying 6^6 by 6. Therefore, it would be the same as 6^6 x 6^1. When you multiply exponents of the same base, you keep the base and just add the exponents to get the final answer. 6 + 1 = 7. So the answer is 6^7

Nathan Schaffert - 4 years, 11 months ago

Is it true that x^x + x^x + x^x +.......+ x^x (lets say until y times of x^x) The solution would be (x)(x^(y+1) May i ask why is the above = 6 x 6^6? Thanks

Sheryl-Lynn Tan - 3 years, 11 months ago

Now I understand! Thank you.

JINJIE Zhang - 5 months, 3 weeks ago

It's may be easier to think in this way that Lets 6=x,

x^6+x^6+x^6+x^6+x^6+x^6 =6x^6 =6.6^6 (x=6) =6^7

Kazal Kazal - 5 years ago

6 6 + 6 6 + 6 6 + 6 6 + 6 6 + 6 6 = 6 a 6^6+6^6+6^6+6^6+6^6+6^6=6^a l o g ( 6 6 ) + l o g ( 6 6 ) + l o g ( 6 6 ) + l o g ( 6 6 ) + l o g ( 6 6 ) + l o g ( 6 6 ) = l o g ( 6 a ) log(6^6)+log(6^6)+log(6^6)+log(6^6)+log(6^6)+log(6^6)=log(6^a) 6 ∗ l o g ( 6 6 ) = a ∗ l o g 6 6*log(6^6)=a*log 6

a = 36 a=36

Sourav Das - 5 years, 6 months ago

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You can't distribute a logarithm in a sum.

Jan Patrick De Jong Yantorno - 5 years, 5 months ago

6 6 + 6 6 + 6 6 + 6 6 + 6 6 + 6 6 = 6 a log ⁡ ( 6 6 + 6 6 + 6 6 + 6 6 + 6 6 + 6 6 ) = log ⁡ 6 a log ⁡ ( 6 6 + 6 6 + 6 6 + 6 6 + 6 6 + 6 6 ) = a log ⁡ 6 { 6 }^{ 6 }+{ 6 }^{ 6 }+{ 6 }^{ 6 }+{ 6 }^{ 6 }+{ 6 }^{ 6 }+{ 6 }^{ 6 }={ 6 }^{ a }\\ \log { \left( { 6 }^{ 6 }+{ 6 }^{ 6 }+{ 6 }^{ 6 }+{ 6 }^{ 6 }+{ 6 }^{ 6 }+{ 6 }^{ 6 } \right) } =\log { { 6 }^{ a } } \\ \log { \left( { 6 }^{ 6 }+{ 6 }^{ 6 }+{ 6 }^{ 6 }+{ 6 }^{ 6 }+{ 6 }^{ 6 }+{ 6 }^{ 6 } \right) } =a\log { 6 }

This is the correct way to distribute the logarithm but you cannot solve this problem this way.

Edit:

You can solve it this way but the involvement of logarithm is actually unnecessary.

log ⁡ ( 6 6 + 6 6 + 6 6 + 6 6 + 6 6 + 6 6 ) = a log ⁡ 6 log ⁡ [ 6 ( 6 6 ) ] = a log ⁡ 6 log ⁡ 6 7 = a log ⁡ 6 7 log ⁡ 6 = a log ⁡ 6 a = 7 \log { \left( { 6 }^{ 6 }+{ 6 }^{ 6 }+{ 6 }^{ 6 }+{ 6 }^{ 6 }+{ 6 }^{ 6 }+{ 6 }^{ 6 } \right) } =a\log { 6 } \\ \log { \left[ 6({ 6 }^{ 6 }) \right] } =a\log { { 6 } } \\ \log { { 6 }^{ 7 } } =a\log { { 6 } } \\ 7\log { { 6 } } =a\log { { 6 } } \\ a=7

Kenneth Choo - 5 years, 4 months ago

That's what I got

Amy Doerr - 5 years, 5 months ago

I used this way in my mind!

Mandeep Singh - 5 years, 5 months ago

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pagal tera question ghalat ha

Usman Abdullah - 5 years, 5 months ago

your method is wrong

Usman Abdullah - 5 years, 5 months ago

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i think where bases are same, power got add, right?

Nida Ahmed - 5 years, 4 months ago

the answer is a=12.

36^6=6^a (6^2)^6=6^a 6^12=6^a => a=12 those who said a=7 are wrong.

numerically:

(36)^6=2,176,782,336 (6)^12= 2,176,782,336 6^7= 279,936

thank you.

Lalo Duron - 5 years, 2 months ago

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6 6 + 6 6 + 6 6 + 6 6 + 6 6 + 6 6 ≠ 36 6 6 6 + 6 6 + 6 6 + 6 6 + 6 6 + 6 6 = 6 ( 6 6 ) ≠ 36 6 { 6 }^{ 6 }+{ 6 }^{ 6 }+{ 6 }^{ 6 }+{ 6 }^{ 6 }+{ 6 }^{ 6 }+{ 6 }^{ 6 }\neq { 36 }^{ 6 }\\ { 6 }^{ 6 }+{ 6 }^{ 6 }+{ 6 }^{ 6 }+{ 6 }^{ 6 }+{ 6 }^{ 6 }+{ 6 }^{ 6 }=6\left( { 6 }^{ 6 } \right) \neq { 36 }^{ 6 }

Kenneth Choo - 5 years, 2 months ago

6^6=279936 279936x6 = 1679616 6^8 = 1679616 Answer: a=8

Niki De Paoli - 5 years, 6 months ago

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Just a little correction 6 6 = 46656 6^6 = 46656

Actually 279936 = 6 7 279936 = 6^7

Aamir Waheed - 5 years, 6 months ago
Leonardo Sipayung
Mar 16, 2015

6 6 + 6 6 + 6 6 + 6 6 + 6 6 + 6 6 = 6 a 6^6 + 6^6 + 6^6 + 6^6 + 6^6 + 6^6 = 6^a 6 6 ( 1 + 1 + 1 + 1 + 1 + 1 ) = 6 a 6^6(1 + 1 + 1 + 1 + 1 + 1) = 6^a 6 6 ( 6 ) = 6 a 6^6(6) = 6^a 6 7 = 6 a 6^7 = 6^a a = 7 a = 7

234 Helpful 2 Interesting 11 Brilliant 1 Confused

It was easy to understand your solution to this problem

sathya damodaran - 5 years, 9 months ago

lieonerdo your solution satisfied me most

Soiyod Sabab - 5 years, 4 months ago

Used the same method. Easiest way.Why do people use complicated logs in this??just simply take common.Best method in math is the easiest method

Muhaummad Umer - 5 years, 5 months ago

I used this method as well. Easiest method.

Dipankan Bandopadhyay - 5 years, 5 months ago

I did it in this way

Nahid Tuli - 5 years, 5 months ago

Your solution well understood..

Tarnue Borbor - 5 years, 1 month ago

I got a=36. I just don’t see how (1+1...) is the same as (6^6+6^6...) makes no sense.

Mike David - 1 year, 8 months ago

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NM I got it. Just confused my rules.

Mike David - 1 year, 8 months ago

6 6 + 6 6 + 6 6 + 6 6 + 6 6 + 6 6 \Large 6^{6}+6^{6}+6^{6}+6^{6}+6^{6}+6^{6} = 6 ( 6 6 ) \Large =6(6^{6}) = 6 6 + 1 \Large =6^{6+1} = 6 7 \Large =6^{7} a is therefore 7 \Large \text{a is therefore 7}

39 Helpful 0 Interesting 0 Brilliant 1 Confused

Finally, one example that I could understood.

Francisco Gallego - 5 years, 4 months ago

Thank you, this I get..

Matt Dunaway - 5 years, 3 months ago

This is the Simplest and Correct Answer

Sarah Louise Jennings - 5 years ago

X^6 + X^6 + X^6 + X^6 +X^6 +X^6 = 6X^6

When x=6 we have 6 (6^6) = 6^1+6 => a=7 , as per exponential law

Abdul Hafiz Azizi - 2 years, 6 months ago
Chelo Norico
Sep 2, 2014

6 * 6^6

The exponents are 1 and 6. Adding them,

6 + 1 = 7

Answer: 6^7

6 Helpful 0 Interesting 0 Brilliant 0 Confused
Austin Griner
Sep 1, 2015

There are six sixths to the sixth (try saying that six times fast). If you had just 6+6+6+6+6+6, it would be 6 6. The same concept applies here, making it (6^6) 6, which ultimately results in 6^7. I have to admit, I am baffled by how long it took me to realize this.

4 Helpful 0 Interesting 0 Brilliant 0 Confused
Gaurav Dey
Dec 15, 2015

6E6=46656 Now 46656[1+1+1+1+1+1]=6^6*6^1=6^7 a=7.....

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Mohammad Khaza
Jun 24, 2017

6^6+6^6+6^6+6^6+6^6+6^6=6^6x6

=6^6x6^1

=6^(6+1)

=6^7

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Given : 6 6 + 6 6 + 6 6 + 6 6 + 6 6 + 6 6 = 6 a 6^{6} + 6^{6} + 6^{6} + 6^{6} + 6^{6} + 6^{6} = 6^{a}

let x = 6 6 6^{6}

then: x + x + x + x + x + x = 6 a x + x + x + x + x + x = 6^{a}

6 x = 6 a 6x = 6^{a}

6 × 6 6 = 6 a 6 \times 6^{6} = 6^{a}

6 7 = 6 a 6^{7} = 6^{a}

7 = a

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Praveen Sridhar
Dec 9, 2015

6^6+6^6+6^6+6^6+6^6+6^6 = 6*6^6 = 6^7

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Raymond Fang
Jan 23, 2021

6 6 + 6 6 + 6 6 + 6 6 + 6 6 + 6 6 = 6 1 × 6 6 = 6 1 + 6 = 6 7 → a = 7 6^6+6^6+6^6+6^6+6^6+6^6=6^1 \times 6^6=6^{1+6}=6^7\rightarrow a=7

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Swayam Acharya
Dec 15, 2020

The number reapets 6 times.... so,6*6^6=6^7.... now 6^7=6^a as the base is same... so a=7

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Tom Wang
Jun 25, 2020

there are 6 6 6 6^6 s, which gives 6 7 6^7 , hence a = 7 a=7

0 Helpful 0 Interesting 0 Brilliant 0 Confused

6 6 + 6 6 + 6 6 + 6 6 + 6 6 + 6 6 = 6 a 6^6 + 6^6 + 6^6 + 6^6 + 6^6 + 6^6 = 6^a

6 × 6 6 = 6 a 6 \times 6^6 = 6^a

6 6 + 1 = 6 a 6^{6+1} = 6^a

6 7 = 6 a 6^7 = 6^a

⇒ a = 7 \Rightarrow \color{#69047E}\boxed{a = 7}

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Oon Han
Dec 9, 2018

Using the product of exponents property, 6 6 + 6 6 + 6 6 + 6 6 + 6 6 + 6 6 = 6 × 6 6 = 6 1 × 6 6 = 6 1 + 6 = 6 7 \begin{aligned} 6^6 + 6^6 + 6^6 + 6^6 + 6^6 + 6^6 &= 6 \times 6^6 \\ &= 6^1 \times 6^6 \\ &= 6^{1 + 6} \\ &= 6^7 \end{aligned} Therefore, a = 7 a = \boxed{7} .

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Gia Hoàng Phạm
Sep 19, 2018

6 6 + 6 6 + 6 6 + 6 6 + 6 6 + 6 6 = 6 × 6 6 = 6 6 + 1 = 6 7 6^6+6^6+6^6+6^6+6^6+6^6=6 \times 6^6=6^{6+1}=6^7

Hence a = 7 a=\boxed{\large{7}}

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Xi Guan
May 11, 2018

6^1+6^1+6^1+6^1+6^1+6^1=6^2

6^2+6^2+6^2+6^2+6^2+6^2=6^3

6^3+6^3+6^3+6^3+6^3+6^3=6^4

6^4+6^4+6^4+6^4+6^4+6^4=6^5 。 。 。

6^n+6^n+6^n+6^n+6^n+6^n=6^n+1

So 6^6+6^6+6^6+6^6+6^6+6^6=6^a, a=6+1=7

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Syed Hamza Khalid
May 19, 2017

6 6 + 6 6 + 6 6 + 6 6 + 6 6 + 6 6 + 6 6 = 6 ∗ 6 6 = 6 7 6^6+6^6+6^6+6^6+6^6+6^6+6^6=6*6^6=6^7

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Anwesha Sinha
May 29, 2016

First, we can take 6^6 common...afterwards, 6^6 (1+1+1+1+1+1) will remain which after solving, will be 6^6 ×6..we know that on multiplication, powers add up..hence, after adding the powers of 6, we get...6^7=6^a. So, a=7

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Muhammed Ameer P
Mar 9, 2016

5+5+5+5+5+5= 5×6=30 Like that 6^6×6^6×6^6×6^6×6^6×6^6=( 6^6)×6=(6^6)×6^1= 6^(6+1)= 6^7

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Kamalpreet Singh
Jan 28, 2016

It was a=7

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Mohit Bhavsar
Jan 6, 2016

6^6(1+1+1+1+1+1) =6^6(6) =6^7 Therefore a=7

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Youssef Mohammed
Jan 5, 2016

6 * 6^6 = 6^7 :))

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Binayak Dey
Dec 23, 2015

6^6+6^6+6^6+6^6+6^6+6^6=6•6^6=6^7 which is equal to 6^a.So a=7

0 Helpful 0 Interesting 0 Brilliant 0 Confused

6^6+6^6+6^6+6^6+6^6+6^6=6^a
=> 6^6 * 6
=> 6^1 + 6^6
=>6^7
a=7



0 Helpful 0 Interesting 0 Brilliant 0 Confused
Ahmad Hassan
Nov 9, 2015

Is that all?

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