GCSE Estimating powers and roots Practice Questions

Find the square numbers either side of $50$

List square numbers until $50$ is trapped: $49$ is $7$ squared and $64$ is $8$ squared.

Take square roots across the bracket

Square rooting keeps the order for positive numbers, so root $50$ stays trapped in the middle.

State the integers

So root $50$ lies between the consecutive integers $7$ and $8$.

Square the whole number

To compare a root with a whole number, square them both. $8$ squared is $64$.

Square the root

Squaring root $60$ gives exactly $60$.

Compare the squares

Since $64$ beats $60$, the number $8$ is greater than root $60$.

Pick a common test value

Both roots are a little above $4$, so test each against the same value, $4.1$.

Test the square root

$4.1$ squared is $16.81$, below $17$ — so root $17$ is above $4.1$.

Test the cube root

$4.1$ cubed is $68.921$, above $65$ — so the cube root of $65$ is below $4.1$.

Put the results together

The same test value $4.1$ sits between the two roots.

Draw the conclusion

So root $17$ is the greater number.

Sense check with decimals

The approximate values $4.12$ and $4.02$ confirm the comparison.

Plan the comparison

All three values are close to $9.9$, so compare each root against carefully chosen decimals.

Compare root $99$ with $9.9$

$9.9$ squared falls short of $99$, so root $99$ is bigger than $9.9$.

Compare the cube root with $9.9$

$9.9$ cubed falls short of $980$, so the cube root of $980$ is also bigger than $9.9$. So $9.9$ is the smallest.

Now separate the two roots

Both roots beat $9.9$, so test a finer value against each.

Test the cube root at $9.94$

$9.94$ cubed is about $982$, above $980$ — so the cube root of $980$ is below $9.94$.

Test root $99$ at $9.94$

$9.94$ squared is $98.80$, below $99$ — so root $99$ is above $9.94$.

Put the roots in order

The same test value $9.94$ separates the two roots.

Assemble the full order

Combining both comparisons gives the complete ordering.

Sense check with decimals

The approximate values confirm the order: $9.9$, then $9.933$, then $9.950$.

State the answer

From smallest to largest: $9.9$, the cube root of $980$, then root $99$.

Understand the condition

We need the smallest integer n whose square root beats 20.5.

Square both sides

Squaring keeps the order for positive numbers, turning the root condition into a plain inequality.

Plan the squaring

Split $20.5$ to square it mentally.

Expand the square

Use the expansion of a two-part square: $400$ plus $20$ plus $0.25$.

Evaluate

So the condition is n greater than 420.25.

Find the smallest integer

The smallest integer strictly greater than $420.25$ is $421$.

Check $421$ works

$421$ is beyond $420.25$, so its root is beyond $20.5$. ✓

Check $420$ fails

$420$ falls short of $420.25$, so its root falls short of $20.5$ — $420$ fails.

Confirm minimality

Since 420 fails and 421 works, 421 is the smallest possible n.

Estimate the root of $421$ as a check

Root $421$ is about $20.518$, just above $20.5$ — a comfortable but tight pass.

Watch for the boundary error

Rounding $420.25$ down to $420$ is the classic slip; the strict inequality demands going above $420.25$.

Watch for the equality trap

Even the exact value 420.25 (if n could be a decimal) only equals 20.5 — the strict inequality rules it out.

Reflect on the method

Threshold problems with roots are solved by squaring the boundary and taking the next integer.

General lesson

For positive values, root conditions convert exactly into squared conditions.

State the answer

The smallest integer n is 421.