# 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.

### The Square of My Age

##### 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?

### Square Triangle

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

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

### Mini Cross-number

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

Which digit replaces x in this crossnumber?

### Maundy Money

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

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

### Age of Augustus

##### 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?

### Tenth Power

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

Do these powers look odd...?

### Square Percentage

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

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

### Not a Zero

##### 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...

### Power up to 1000

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

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

### The Square and the Root

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

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

### Root 2017

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

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

### Powers of Four

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

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

### The Power of the Sum

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

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

### Megabytes and Kilobytes

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

How many bytes are there in one megabyte?

### Largest Expression

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

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

### Roots Near 9

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

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

### Which Power?

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

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

### The Power of $x$

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

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

### Cube Factors

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

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

### Two in a Million

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

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

### How Many Squares?

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

How many 4-digit numbers are perfect squares?

### Huge Powers

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

Can you put these numbers in order?

### Powerful Expressions

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

Put these expressions in order, from smallest to largest.

### Root Estimation

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

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

### Powerful Order

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

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

### Rooted Via 10

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

How many of the numbers shown are greater than 10?

### Rough Root

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

Which of these options is closest to this square root?

### Doubly Powerful

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

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

### Big Product

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

How many digits are there in this product?

### Powerful 9

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

What is the last digit of this calculation?

### Power of 3

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

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

### Powered Up

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

Can you work out the product of these indices?

### Power of Five

##### 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 $xy$?

### Self-power Squares

##### 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?

### Great Power

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

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