Without using a calculator how can we say 6^20> 3.10^8?

Question

Please help me understand the method you are using.

score 4 · Answer 1 · answered Feb 27 '18 at 06:01

4

$6^{20}=6^{4}6^{8}6^{8}>6^{4}2^{8}5^{8}=6^{4}10^{8}>(3)10^8$

Also, if there were meant to be parentheses: $$6^{20}=(6^2)^{10}=36^{10}>30^{10}>30^{8}$$

answered Feb 27 '18 at 06:01

Chris

384

Are you familiar with the property of exponents: $a^{b+c}=a^{b}a^{c}$? – Chris Feb 27 '18 at 06:09
Yes,I am femiliar with the property sir. – Mervin Jacob Feb 27 '18 at 06:12

user · Answer 2 · 2018-02-27T06:31:45.163

2

Note that

$$2^{10}=1024>1000=10^3$$

then

$$6^{20}=3^{20}2^{20}>2^{20}2^{20}=2^{40}=(2^{10})^4>(10^3)^4=10^{12}$$

edited Feb 27 '18 at 06:31

answered Feb 27 '18 at 06:24

user

154,566

Sir,How did this come " (10^3)^4 "? – Mervin Jacob Feb 27 '18 at 06:30
in the first line we have shown that 2^{10}>10^3 then we can sustitute – user Feb 27 '18 at 06:32
Ok sir , now I understood. – Mervin Jacob Feb 27 '18 at 06:36
We’ll done, Bye. – user Feb 27 '18 at 06:53

score 2 · Accepted Answer · answered Feb 27 '18 at 07:31

2

And one more :

$6^{20}=(6^2)^{10}= (36)^{10} > (3 \cdot 10)^{10}=$

$3^{10}10^{10} >3×10^8.$

answered Feb 27 '18 at 07:31

Peter Szilas

20,344
2
17
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Nice Peter, very straightforward. – user Feb 27 '18 at 07:42
1

Gimusi. Thanks, not so very difficult:)) – Peter Szilas Feb 27 '18 at 10:23
Brevity and clearness, very nice! – user Feb 27 '18 at 10:24

score 0 · Answer 4 · 2018-02-27T07:34:17.640

0

$$6^{20}=(2\cdot3)^{20}=2^{20}\cdot 3^{20}=2^{20}\cdot {(3^5)}^4=2^{20}\cdot {243}^4\gt2^{20}\cdot {200}^4=2^{20}\cdot (2\cdot 100)^4=2^{20}\cdot 2^4\cdot 100^4=2^{24}\cdot {(10^2)}^4=2^{24}\cdot 10^8\gt3\cdot 10^8$$

edited Feb 27 '18 at 07:34

answered Feb 27 '18 at 07:06

Without using a calculator how can we say 6^20> 3.10^8?

4 Answers4