List

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
problem

The square of my age

Age
11 to 14
Challenge level
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Lauren and Thomas tell their ages in terms of sums of squares. Can you work out how old they really are?
Mini Cross-Number
problem

Mini cross-number

Age
11 to 14
Challenge level
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Which digit replaces x in this crossnumber?
Square Percentage
problem

Square percentage

Age
11 to 14
Challenge level
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What percentage of the integers between 1 and 10,000 are square numbers?
Tenth Power
problem

Tenth power

Age
11 to 14
Challenge level
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Do these powers look odd...?
Square Triangle
problem

Square triangle

Age
11 to 14
Challenge level
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How many triangles have all three angles perfect squares (in degrees)?
Maundy Money
problem

Maundy money

Age
11 to 14
Challenge level
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How much money did the Queen give away in pence as a power of 2?
Age of Augustus
problem

Age of augustus

Age
11 to 14
Challenge level
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The English mathematician Augustus de Morgan has given his age in algebraic terms. Can you work out when he was born?
Place value, integers, ordering and rounding - Short Problems
problem

Root 2017

Age
14 to 16
Challenge level
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Between which two whole numbers is $\sqrt{2017}$ $?$

Place value, integers, ordering and rounding - Short Problems
problem

The square and the root

Age
14 to 16
Challenge level
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What is the remainder when the square of 49 is divided by the square root of 49?

Place value, integers, ordering and rounding - Short Problems
problem

The power of the sum

Age
14 to 16
Challenge level
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What is this sum, expressed as a power of 2?

Place value, integers, ordering and rounding - Short Problems
problem

Power up to 1000

Age
14 to 16
Challenge level
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Find $x+y$, where $2^x\times5^y=1000$

Powers of Four
problem

Powers of four

Age
14 to 16
Challenge level
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Can you work out the value of x in this 'power-full' equation?
Pythagorean Quadruple
problem

Pythagorean quadruple

Age
14 to 16
Challenge level
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The sum of three square numbers equals $121$. What can those numbers be...
Doubly Powerful
problem

Doubly powerful

Age
14 to 16
Challenge level
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To what power should $4^4$ be raised to give $8^8$?
Huge powers
problem

Huge powers

Age
14 to 16
Challenge level
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Can you put these numbers in order?
Place value, integers, ordering and rounding - Short Problems
problem

The power of x

Age
14 to 16
Challenge level
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Find the value of $x$ in this equation, where it appears in powers.

Place value, integers, ordering and rounding - Short Problems
problem

Cube factors

Age
14 to 16
Challenge level
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How many factors of $9^9$ are perfect cubes?

Place value, integers, ordering and rounding - Short Problems
problem

How many squares?

Age
14 to 16
Challenge level
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How many 4-digit numbers are perfect squares?

Place value, integers, ordering and rounding - Short Problems
problem

Root estimation

Age
14 to 16
Challenge level
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Which of these is the best approximation for this square root?

Place value, integers, ordering and rounding - Short Problems
problem

Big product

Age
14 to 16
Challenge level
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How many digits are there in this product?

Place value, integers, ordering and rounding - Short Problems
problem

Which power?

Age
14 to 16
Challenge level
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Which power of $16$ is equal to $64^6$?

Place value, integers, ordering and rounding - Short Problems
problem

Two in a million

Age
14 to 16
Challenge level
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What is the highest power of 2 that divides exactly into 1000000?

Place value, integers, ordering and rounding - Short Problems
problem

Powerful expressions

Age
14 to 16
Challenge level
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Put these expressions in order, from smallest to largest.

Place value, integers, ordering and rounding - Short Problems
problem

Rough root

Age
14 to 16
Challenge level
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Which of these options is closest to this square root?

Rooted via 10
problem

Rooted via 10

Age
14 to 16
Challenge level
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How many of the numbers shown are greater than 10?
Largest Expression
problem

Largest expression

Age
14 to 16
Challenge level
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Which of these five algebraic expressions is largest, given $x$ is between 0 and 1?
Place value, integers, ordering and rounding - Short Problems
problem

Roots near 9

Age
14 to 16
Challenge level
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For how many integers n is the difference between √n and 9 less than 1?

Powerful Order
problem

Powerful order

Age
14 to 16
Challenge level
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Powers of numbers might look large, but which of these is the largest...
Powered Up
problem

Powered up

Age
14 to 16
Challenge level
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Can you work out the product of these indices?
Self-Power Squares
problem

Self-power squares

Age
14 to 16
Challenge level
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How many integers $n$, between $1$ and $100$ inclusive, have the property that $n^n$ is a square number?
Power of 3
problem

Power of 3

Age
14 to 16
Challenge level
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What power of 27 is needed to get the correct power of 3?
Know Your Powers
problem

Know your powers

Age
14 to 16
Challenge level
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Given these equations with unknown powers $x$ and $y$, can you work out $xy$?
Place value, integers, ordering and rounding - Short Problems
problem

Great power

Age
14 to 16
Challenge level
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Which is greater: $10^{250}$ or $6^{300}$?

Place value, integers, ordering and rounding - Short Problems
problem

Powerful 9

Age
14 to 16
Challenge level
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What is the last digit of this calculation?

Place value, integers, ordering and rounding - Short Problems
problem

Power of five

Age
14 to 16
Challenge level
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Powers with brackets, addition and multiplication