Large exponents how many digits in 2^2009×5^1957÷4^24

[mathhelpplease14]

Large exponents how many digits in 2^2009×5^1957÷4^24

how many digits are in the number?
2^2009×5^1957÷4^24

Please help I know nothing of logarithiums and have no been taught yet so please do not answer using them. I think it has something to do with the 2×5=10 and 10 with exponents is easy but really I have no idea how to answer this question please help.

 

[royhaas]

Re: Large exponents how many digits

Start by writing \(\displaystyle 4^{24}=2^{48}\). Now use the laws of exponents.

 

[galactus]

Re: Large exponents how many digits

If you know nothing of logs, then now is the time to learn. They are handy with huge numbers like these. That is why they were used in the first place.

Here is a generality for you. Suppose we want to find out how many digits are in

\(\displaystyle 2^{123456}\)

That is a huge number. But, if we take \(\displaystyle log(2^{123456})=123456log(2)=37163.9591447\)

Round up and there are 37164 digits in that huge number.

To see what those first several digits are, take 10 to the power of the decimal part of the above.

\(\displaystyle 10^{.9591447}=9.10216491744\)

The number begins \(\displaystyle 910216491744..................\text{37151 digits}...............2\)

 

[soroban]

Re: Large exponents how many digits in 2^2009×5^1957÷4^24

Hello, "mathhelpplease14!

\(\displaystyle \text{How many digits are in the number: }\:\frac{2^{2009}\cdot5^{1957}}{4^{24}}\)

You're right . . . all thosr 2's and 5's are big clue . . .


\(\displaystyle \text{Note that: }\:4^{24} \:=\2^2)^{24} \:=\:2^{48}\)

\(\displaystyle \text{So we have: }\:\frac{2^{2009}\cdot5^{1957}}{2^{48}} \;=\;2^{1961}\cdot5^{1957} \;=\;2^4\cdot2^{1957}\cdot5^{1957}\)

. . . . \(\displaystyle \;=\;2^4\cdot\left(2\cdot5)^{1957} \;=\;2^4\cdot10^{1957}\)


\(\displaystyle \text{So, the number has "16", followed by 1957 zeros.}\)

. . \(\displaystyle \text{Therefore, it has 1959 digits.}\)