Square Power
What is square root of 5 multiplied by 2, to the third power?
What is the square root of 5, multiplied by 2, to the third power.
Since my math course is a course that prohibits the use of a calculator, I’d like an answer resulted without using a calculator.
It seems unlikely anyone would expect you to work out the square root of five without a calculator. Assuming they do however, you need to use newton’s method.
See:
http://planetmath.org/encyclopedia/NewtonRaphsonMethod.html
So, we know that
√4 < √5 < √9 thus 2 < √5 < 3
So we pick a reasonable number, say 2.5
You then use an interative process using the newton formule, I will use a and b for a1 and a0 for ease of display.
so b is your next estimate and a is your current estimate. (if you are looking at the website you will notice we use minus 5 in our example as this is the value we are trying to get the root of)
b = a - [(a^2 - 5)/2a)
for our example, b = 2.5 - [(2.5^2 - 5)/5] = 2.5 - (1.25/5) = 2.25
so now b= 2.25 - [(2.25^2 - 5)/4.5] = 2.25 - (0.0625/4.5) = 2.236
Since the modifier was so small (even after just two iterations) we can be pretty sure this is very close to the true value.
No we just multiply by 2 = 4.472
to the power of 3 = 4.472 x 4.472 x 4.472 = 89.43.
Labor intensive - but can be done without a calculator!
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