# Calculate fractional exponent without use of calculator

Let we have a formula to calculate Effort.

Effort=$a \times (\text{KLOC})^{b}$

where $a=2.8,b=1.20,\text{KLOC=40}$

So Effort$=2.8 \times (\text{40})^{1.20}$

Options are-:

A. $234.25$

B. $932.50$

C. $287.80$

D. $122.40$

## My doubt/problem

I have to solve the value of Effort wihtout using Calculator.I have no clue how to solve it.

Any hint will be appreciated.Thanks

Note that $$(40)^{1.2} = 40^{1 + 0.2} = 40^1 \times 40^{0.2} = 40 \times \sqrt[5]{40}$$ Now $2^5 = 32$ while $3^5 = 243$ (you don't have to remember these - they can be calculated fairly quickly). So $40$ is going to be a 5th power of something just a little over $2$. So that gives us $$2.8 \times 40 \times (2 + x)$$for some small number $x$. If we increase the $2.8$ to $3$ and discard the $x$, we get $$3 \times 40 \times 2 = 240$$ Since $x$ is almost certainly smaller than $0.2$, it is reasonable to expect that increasing $2.8$ to $3$ is going to have a bigger impact than decreasing $2 + x$ to $2$. Thus we should expect the answer to be close to but a little smaller than $240$.
So the answer should be A: $234.25$.