# Powers and Roots - Short Problems

This is part of our collection of

Short Problems.

You may also be interested in our longer problems on

Powers and Roots.

Printable worksheets containing selections of these problems are available

here.

##### Age 11 to 14 Short Challenge Level:

Lauren and Thomas tell their ages in terms of sums of squares. Can you work out how old they really are?

##### Age 11 to 14 Short Challenge Level:

How many triangles have all three angles perfect squares (in degrees)?

##### Age 11 to 14 Short Challenge Level:

Which digit replaces x in this crossnumber?

##### Age 11 to 14 Short Challenge Level:

How much money did the Queen give away in pence as a power of 2?

##### Age 11 to 14 Short Challenge Level:

The English mathematician Augustus de Morgan has given his age in algebraic terms. Can you work out when he was born?

##### Age 11 to 14 Short Challenge Level:

Do these powers look odd...?

##### Age 11 to 14 Short Challenge Level:

What percentage of the integers between 1 and 10,000 are square numbers?

##### Age 11 to 14 Short Challenge Level:

What is the last non-zero digit of $2^{57} \times 3^4 \times 5^{53}$?

##### Age 14 to 16 Short Challenge Level:

The sum of three square numbers equals $121$. What can those numbers be...

##### Age 14 to 16 Short Challenge Level:

Find $x+y$, where $2^x\times5^y=1000$

##### Age 14 to 16 Short Challenge Level:

What is the remainder when the square of 49 is divided by the square root of 49?

##### Age 14 to 16 Short Challenge Level:

Between which two whole numbers is $\sqrt{2017}$ $?$

##### Age 14 to 16 Short Challenge Level:

Can you work out the value of x in this 'power-full' equation?

##### Age 14 to 16 Short Challenge Level:

What is this sum, expressed as a power of 2?

##### Age 14 to 16 Short Challenge Level:

How many bytes are there in one megabyte?

##### Age 14 to 16 Short Challenge Level:

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

##### Age 14 to 16 Short Challenge Level:

For how many integers 𝑛 is the difference between √𝑛 and 9 is less than 1?

##### Age 14 to 16 Short Challenge Level:

Which power of $16$ is equal to $64^6$?

##### Age 14 to 16 Short Challenge Level:

Find the value of $x$ in this equation, where it appears in powers.

##### Age 14 to 16 Short Challenge Level:

How many factors of $9^9$ are perfect cubes?

##### Age 14 to 16 Short Challenge Level:

What is the highest power of 2 that divides exactly into 1000000?

##### Age 14 to 16 Short Challenge Level:

How many 4-digit numbers are perfect squares?

##### Age 14 to 16 Short Challenge Level:

Can you put these numbers in order?

##### Age 14 to 16 Short Challenge Level:

Put these expressions in order, from smallest to largest.

##### Age 14 to 16 Short Challenge Level:

Which of these is the best approximation for this square root?

##### Age 14 to 16 Short Challenge Level:

Powers of numbers might look large, but which of these is the largest...

##### Age 14 to 16 Short Challenge Level:

How many of the numbers shown are greater than 10?

##### Age 14 to 16 Short Challenge Level:

Which of these options is closest to this square root?

##### Age 14 to 16 Short Challenge Level:

To what power should $4^4$ be raised to give $8^8$?

##### Age 14 to 16 Short Challenge Level:

How many digits are there in this product?

##### Age 14 to 16 Short Challenge Level:

What is the last digit of this calculation?

##### Age 14 to 16 Short Challenge Level:

What power of 27 is needed to get the correct power of 3?

##### Age 14 to 16 Short Challenge Level:

Can you work out the product of these indices?

##### Age 14 to 16 Short Challenge Level:

Powers with brackets, addition and multiplication

##### Age 14 to 16 Short Challenge Level:

Given these equations with unknown powers $x$ and $y$, can you work out $xy$?

##### Age 14 to 16 Short Challenge Level:

How many integers $n$, between $1$ and $100$ inclusive, have the property that $n^n$ is a square number?

##### Age 14 to 16 Short Challenge Level:

Which is greater: $10^{250}$ or $6^{300}$?