indice thing 2

red and white kop!

Junior Member
Joined
Jun 15, 2009
Messages
231
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
 
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:

. . [1]    3236    4×10112\displaystyle [1]\;\;3^{236} \;\approx\; 4 \times 10^{112}

. . [2]    3376    4×10180\displaystyle [2]\;\;3^{-376} \;\approx\; 4 \times 10^{-180}

find approximations, also in standard form, for:

(a)  3376\displaystyle (a)\;3^{376}

This is the reciprocal of [2].

3376  =  13376  =  14×10180  =  14×10180  =  0.25×10180  =  2.5×10179\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}




(b)  3612\displaystyle (b)\; 3^{612}

This is [1] divided by [2].

3612  =  32363376  =  4×101124×10180  =  10292\displaystyle 3^{612} \;=\;\frac{3^{236}} {3^{-376}} \;=\; \frac{4\times10^{112}}{4\times10^{-180}} \;=\;10^{292}




(c)  (3)236\displaystyle (c)\; \left(\sqrt3\right)^{236}

(3)236  =  (312)236  =  (3236)12  =  (4×10112)12\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}}

. . . . . . .=  412×(10112)12  =  2×1056\displaystyle =\;4^{\frac{1}{2}} \times \left(10^{112}\right)^{\frac{1}{2}} \;=\;2 \times 10^{56}




(d)  (3376)52\displaystyle (d)\; \left(3^{-376}\right)^{\frac{5}{2}}

(3376)52  =  (4×10180)52  =  452×(10180)52\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}}

. . . . . . .=  32×10450  =  3.2×10449\displaystyle =\;32 \times 10^{-450} \;=\; 3.2 \times 10^{-449}

 
Top