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Big Powers

Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.

Rooted Via 10

How many of the numbers shown are greater than 10?

Largest Expression

Which of these five algebraic expressions is largest, given $x$ is between 0 and 1?

Root Estimation

Age 14 to 16 Short
Challenge Level

Answer: 0.45

Estimating the root
$10000=100^2$, so $\dfrac{2001}{10000} \approx \left(\dfrac{?}{100}\right)^2$
$?^2\approx 2001$
$40^2=1600$ too small
$50^2=2500$ too big
$45^2=2025$ close but too big
$44^2=1936$ close but not as close
So $\dfrac{2001}{10000}\approx\left(\dfrac{45}{100}\right)^2$
$\therefore \sqrt{\dfrac{2001}{10000}}\approx\dfrac{45}{100}=0.45$

Squaring the options

$0.4^2=0.16$ too small
$0.42^2=0.1764$ too small
$0.45^2=0.2025$ too big
So $0.47^2$ will be too big too

$\sqrt{0.2001}$ is between $0.42$ and $0.45$. We should check numbers between $0.42$ and $0.45$ to find whether it is closer to $0.42$ or $0.45$

$0.2025$ closer than $0.1764$ to suspect $0.45$ closer so try $0.44$:
$0.44^2=0.1936$ too small
So $\sqrt{0.2001}$ is between $0.44$ and $0.45$
The option is it closest to is $0.45$

You can find more short problems, arranged by curriculum topic, in our short problems collection.