Powers and Roots Short Problems: Age 11-16

This is part of our collection of Short Problems.

You may also be interested in our longer problems on Powers and Roots: Age 11-16.

We have put together a selection of these short problems as printable worksheets:

Printable worksheet: Age 11-14 ⭐⭐   Printable worksheet: Age 14-16 ⭐   Printable worksheets: Age 14-16 ⭐⭐   Printable worksheet: Age 14-16 ⭐⭐⭐

Printable worksheet solutions


  • Small tomato seedlings in pink pots.
    problem

    Tenth Power

    Age
    11 to 14
    Challenge level

    Do these powers look odd...?

  • Small tomato seedlings in pink pots.
    problem

    Square Triangle

    Age
    11 to 14
    Challenge level

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

  • Small tomato seedlings in pink pots.
    problem

    Maundy Money

    Age
    11 to 14
    Challenge level

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

  • Small tomato seedlings in pink pots.
    problem

    Age of Augustus

    Age
    11 to 14
    Challenge level

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

  • Small tomato seedlings in pink pots.
    problem

    The Square of My Age

    Age
    11 to 14
    Challenge level

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

  • Small tomato seedlings in pink pots.
    problem

    Mini Cross-Number

    Age
    11 to 14
    Challenge level

    Which digit replaces x in this crossnumber?

  • Small tomato seedlings in pink pots.
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    Square Percentage

    Age
    11 to 14
    Challenge level

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

  • Small tomato seedlings in pink pots.
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    Powers of Four

    Age
    14 to 16
    Challenge level

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

  • Small tomato seedlings in pink pots.
    problem

    Pythagorean Quadruple

    Age
    14 to 16
    Challenge level

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

  • Small tomato seedlings in pink pots.
    problem

    Megabytes and Kilobytes

    Age
    14 to 16
    Challenge level

    How many bytes are there in one megabyte?

  • Small tomato seedlings in pink pots.
    problem

    Root 2017

    Age
    14 to 16
    Challenge level

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

  • Small tomato seedlings in pink pots.
    problem

    The Square and the Root

    Age
    14 to 16
    Challenge level

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

  • Small tomato seedlings in pink pots.
    problem

    The Power of the Sum

    Age
    14 to 16
    Challenge level

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

  • Small tomato seedlings in pink pots.
    problem

    Power Up to 1000

    Age
    14 to 16
    Challenge level

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

  • Small tomato seedlings in pink pots.
    problem

    Rooted Via 10

    Age
    14 to 16
    Challenge level

    How many of the numbers shown are greater than 10?

  • Small tomato seedlings in pink pots.
    problem

    Largest Expression

    Age
    14 to 16
    Challenge level

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

  • Small tomato seedlings in pink pots.
    problem

    Powerful Order

    Age
    14 to 16
    Challenge level

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

  • Small tomato seedlings in pink pots.
    problem

    Doubly Powerful

    Age
    14 to 16
    Challenge level

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

  • Small tomato seedlings in pink pots.
    problem

    Huge Powers

    Age
    14 to 16
    Challenge level

    Can you put these numbers in order?

  • Small tomato seedlings in pink pots.
    problem

    The Power of X

    Age
    14 to 16
    Challenge level

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

  • Small tomato seedlings in pink pots.
    problem

    Cube Factors

    Age
    14 to 16
    Challenge level

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

  • Small tomato seedlings in pink pots.
    problem

    How Many Squares?

    Age
    14 to 16
    Challenge level

    How many 4-digit numbers are perfect squares?

  • Small tomato seedlings in pink pots.
    problem

    Root Estimation

    Age
    14 to 16
    Challenge level

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

  • Small tomato seedlings in pink pots.
    problem

    Big Product

    Age
    14 to 16
    Challenge level

    How many digits are there in this product?

  • Small tomato seedlings in pink pots.
    problem

    Which Power?

    Age
    14 to 16
    Challenge level

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

  • Small tomato seedlings in pink pots.
    problem

    Two in a Million

    Age
    14 to 16
    Challenge level

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

  • Small tomato seedlings in pink pots.
    problem

    Powerful Expressions

    Age
    14 to 16
    Challenge level

    Put these expressions in order, from smallest to largest.

  • Small tomato seedlings in pink pots.
    problem

    Rough Root

    Age
    14 to 16
    Challenge level

    Which of these options is closest to this square root?

  • Small tomato seedlings in pink pots.
    problem

    Roots Near 9

    Age
    14 to 16
    Challenge level

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

  • Small tomato seedlings in pink pots.
    problem

    Powered Up

    Age
    14 to 16
    Challenge level

    Can you work out the product of these indices?

  • Small tomato seedlings in pink pots.
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    Self-Power Squares

    Age
    14 to 16
    Challenge level

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

  • Small tomato seedlings in pink pots.
    problem

    Power of 3

    Age
    14 to 16
    Challenge level

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

  • Small tomato seedlings in pink pots.
    problem

    Know Your Powers

    Age
    14 to 16
    Challenge level

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

  • Small tomato seedlings in pink pots.
    problem

    Great Power

    Age
    14 to 16
    Challenge level

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

  • Small tomato seedlings in pink pots.
    problem

    Powerful 9

    Age
    14 to 16
    Challenge level

    What is the last digit of this calculation?

  • Small tomato seedlings in pink pots.
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    Power of Five

    Age
    14 to 16
    Challenge level

    Powers with brackets, addition and multiplication

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