Hard GCSE Estimating powers and roots Questions

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.

State the known fact

The given square is close to $9750$, so root $9750$ is close to $99$.

Compare with the target

$9750$ is below $99$ squared, so its root is below $99$.

Find the square below

$98$ squared is $9604$, below $9750$.

Bracket the root

The root is trapped between $98$ and $99$.

Plan the midpoint test

Compare the root with $98.5$ to decide the nearest integer.

Square the midpoint

$98.5$ squared is $9702.25$.

Compare with $9750$

$9750$ is beyond the midpoint square.

Interpret

The root is in the upper half of the gap — closer to $99$.

Refine with $98.7$

$98.7$ squared is $9741.69$, still below $9750$ — the root is above $98.7$.

Refine with $98.8$

$98.8$ squared is $9761.44$, above $9750$ — the root is below $98.8$.

Locate the root

So the root is about $98.7$ or so — clearly nearer $99$ than $98$.

Use the gap sizes as a check

$9750$ hugs the $9801$ end of the bracket, agreeing that the root is close to $99$.

Round to the nearest integer

Being above $98.5$, the root rounds to $99$.

Sense check

The true value is about $98.74$, which indeed rounds to $99$. ✓

State the answer

The integer closest to root $9750$ is $99$.

Understand the equation

We need the positive fourth root of $50$, bracketed between consecutive tenths.

Plan via repeated squaring

A fourth power is a square of a square, so trial values can be squared twice.

Bracket with whole numbers

50 sits between 16 and 81, so x is between 2 and 3.

Judge where in the gap

Since x squared is root 50, about 7.07, x is around root 7.07 — roughly 2.66. Start trials near there.

Try $2.6$: square once

First squaring of $2.6$.

Square again

2.6 to the fourth is 45.6976, below 50 — so x is above 2.6.

Try $2.7$: square once

First squaring of $2.7$.

Square again

2.7 to the fourth is 53.1441, above 50 — so x is below 2.7.

State the bracket

x is trapped between the consecutive tenths 2.6 and 2.7.

Verify the squares used

Checking: $676$ squared is $456976$, so $6.76$ squared is $45.6976$ as claimed.

Verify the other square

$729$ squared is $531441$, so $7.29$ squared is $53.1441$ as claimed.

Sense check

The true fourth root of $50$ is about $2.66$, inside the bracket. ✓

Connect to fractional indices

The fourth root is the square root taken twice — the same structure as the repeated squaring used in the trials.

Reflect on the method

Squaring twice keeps each trial to two easy multiplications.

State the answer

x lies between 2.6 and 2.7.

State the student's method

The student averaged the roots of $36$ and $100$ because $60$ lies between them.

Test the answer

If $8$ were close to root $60$, then $8$ squared should be close to $60$ — but it is $64$, noticeably off.

Identify the first flaw

$60$ is not midway between $36$ and $100$ — the midpoint of those squares is $68$, so the average is anchored to the wrong place.

Identify the deeper flaw

Even at the exact midpoint, averaging roots overestimates: the square root graph curves, so the straight-line average sits above it.

See the flaw numerically

Root $68$ is about $8.25$ — and root $60$ must be smaller still, so $8$ cannot be right for root $60$.

Start the correct method

Bracket $60$ with the nearest squares: $7$ squared and $8$ squared.

Bracket the root

Root 60 is between 7 and 8 — already the student's 8 is exposed as an endpoint, not an estimate.

Try $7.7$

$7.7$ squared is $59.29$, just below $60$.

Try $7.8$

$7.8$ squared is $60.84$, just above $60$.

Bracket to tenths

Root $60$ is trapped between $7.7$ and $7.8$.

Test the halfway value

$7.75$ squared is just above $60$, so root $60$ is just below $7.75$.

Round to one decimal place

Being below $7.75$, root $60$ rounds to $7.7$.

Compare the two estimates

The trial-squares estimate of 7.7 is far more accurate than the student's 8 — the true value is about 7.746.

Summarise the error

In one sentence: averaging the roots of two convenient squares is not a valid way to estimate a root in between.

State the better estimate

A better estimate of root $60$ is $7.7$ to one decimal place.

State the known fact

We must stretch this single fact to estimate the root of $170000$.

Scale the fact by $100$

Multiplying $41$ by $10$ multiplies its square by $100$, giving a square very close to $170000$.

Compare with the target

$170000$ is a little beyond $410$ squared, so its root is a little beyond $410$.

Gauge the gap

The target is $1900$ above the known square.

Estimate the step per unit

Each step of $1$ in the root adds about $820$ to the square at this size (the two roots sum to $821$).

Estimate the extra distance

$1900$ divided by roughly $820$ suggests the root is about $2.3$ above $410$.

Form the candidate

So the root should be near $412$, and we check the neighbouring integers.

Check $412$

$412$ squared is $169744$, below $170000$.

Check $413$

$413$ squared is $170569$, above $170000$.

Bracket the root

The root is trapped between $412$ and $413$.

Test the midpoint

The midpoint squares to just above $170000$.

Interpret the test

The root is below $412.5$, so it rounds down to $412$.

Round to the nearest integer

To the nearest integer, the root is $412$.

Sense check

The true value is about $412.3$ — matching the step-size estimate from earlier. ✓

State the answer

The nearest integer to root $170000$ is $412$.