SOLUTION: Estimate 247^9 without using a calculator. I asked my teacher and she said a hint was 200 < 247 < 300. I'm still really stuck. Please help?

Question 1209409: Estimate 247^9 without using a calculator. I asked my teacher and she said a hint was 200 < 247 < 300. I'm still really stuck. Please help?
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!

200 < 247 < 300
200⁹ < 247⁹ < 300⁹

Raising each part to the 9^th power will not flip the inequality signs.
To estimate 247⁹, we need to compute 200⁹ and 300⁹
Which in turn means we need 2⁹ = 512 and 3⁹ = 19683
Both of which can be computed without a calculator when given enough time.

200⁹ = (2*10²)⁹
200⁹ = 2⁹*(10²)⁹
200⁹ = 512*10^2*9
200⁹ = 512*10¹⁸
200⁹ = 5.12*10²*10¹⁸
200⁹ = 5.12*10²⁺¹⁸
200⁹ = 5.12*10²⁰
This massive number is 512 followed by 18 zeros.

300⁹ = (3*10²)⁹
300⁹ = 3⁹*(10²)⁹
300⁹ = 19683*10¹⁸
300⁹ = 1.9683*10⁴*10¹⁸
300⁹ = 1.9683*10⁴⁺¹⁸
300⁹ = 1.9683*10²²
This massive number is 19683 followed by 18 zeros.

We go from this
200⁹ < 247⁹ < 300⁹
to this
5.12*10²⁰ < 247⁹ < 1.9683*10²²
Since the numbers involved are so large, I recommend to keep things in scientific notation.

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