Exponents numbers

Jan 2008
173
1
if 2^5 = 32 then 2^100 is close to

a 10^10 b 10^15 c 10^10 d 10^25 e 10^30
 

Soroban

MHF Hall of Honor
May 2006
12,028
6,341
Lexington, MA (USA)
Hello, sri340!

If \(\displaystyle 2^5 = 32\), then \(\displaystyle 2^{100}\) is close to:

\(\displaystyle (a)\;10^{10} \qquad(b)\;10^{15} \qquad(c)\;10^{10} \qquad(d)\;10^{25} \qquad(e)\;10^{30}\)

Given: .\(\displaystyle 2^5 \:=\:32\)

Square: .\(\displaystyle \left(2^5\right)^2 \:=\:32^2 \quad\Rightarrow\quad 2^{10} \:=\:1024\)

. . . . . . .\(\displaystyle 2^{10} \;\approx\;1000 \quad\Rightarrow\quad 2^{10} \;\approx\;10^3\)


Raise to the 10th power: .\(\displaystyle \left(2^{10}\right)^{10} \:\approx\:\left(10^3\right)^{10}\)

Therefore: . \(\displaystyle 2^{100} \;\approx\;10^{30}\) . . . answer (e)