indice thing 2

[red and white kop!]

given that, in standard form:
3^236 is approximately equal to 4 x 10^112
3^-376 is approximately equal to 4 x 10^-180
find approximations, also in standard form, for
a. 3^376
b. 3^612
c. (sqrt3)^236
d. (3^-376)^(5/2)

i dont know how to apply the given info to the question! i dont even know where to start. this must be simple when its understood, so i just need a helpful hint

 

[soroban]

Hello, red and white kop!

Try to express the number in terms of the given quantities.
It takes a bit of Imagination.

Given that, in standard form:

. . \(\displaystyle [1]\;\;3^{236} \;\approx\; 4 \times 10^{112}\)

. . \(\displaystyle [2]\;\;3^{-376} \;\approx\; 4 \times 10^{-180}\)

find approximations, also in standard form, for:

\(\displaystyle (a)\;3^{376}\)

This is the reciprocal of [2].

\(\displaystyle 3^{376} \;=\;\frac{1}{3^{-376}} \;=\;\frac{1}{4\times 10^{-180}} \;=\;\frac{1}{4} \times 10^{180} \;=\; 0.25 \times 10^{180} \;=\;2.5 \times 10^{179}\)

\(\displaystyle (b)\; 3^{612}\)

This is [1] divided by [2].

\(\displaystyle 3^{612} \;=\;\frac{3^{236}} {3^{-376}} \;=\; \frac{4\times10^{112}}{4\times10^{-180}} \;=\;10^{292}\)

\(\displaystyle (c)\; \left(\sqrt3\right)^{236}\)

\(\displaystyle \left(\sqrt{3}\right)^{236} \;=\;\left(3^{\frac{1}{2}}\right)^{236} \;=\;\left(3^{236} \right)^{\frac{1}{2}} \;=\; \left(4\times 10^{112}\right)^{\frac{1}{2}}\)

. . . . . . .\(\displaystyle =\;4^{\frac{1}{2}} \times \left(10^{112}\right)^{\frac{1}{2}} \;=\;2 \times 10^{56}\)

\(\displaystyle (d)\; \left(3^{-376}\right)^{\frac{5}{2}}\)

\(\displaystyle \left(3^{-376}\right)^{\frac{5}{2}} \;=\;\left(4\times 10^{-180}\right)^{\frac{5}{2}} \;=\; 4^{\frac{5}{2}} \times \left(10^{-180}\right)^{\frac{5}{2}}\)

. . . . . . .\(\displaystyle =\;32 \times 10^{-450} \;=\; 3.2 \times 10^{-449}\)

 

[red and white kop!]

thanks