Hard GCSE Powers and roots Questions

Link tiles to a square array

A square arrangement means side x side tiles.

Find the side

Take the square root of $196$.

Try a number just below

Testing $13$ gives $169$, which is too small.

Try the next number

Testing $14$ gives exactly $196$, so this is the root.

Verify by multiplying back

Check the root by squaring it back to $196$.

Split the second number by place value

Break $14$ into $10 + 4$ so we can multiply in easy parts.

Multiply the tens part

First multiply by the $10$.

Multiply the units part

Then multiply by the $4$.

Add the two parts

Add the partial products to get $196$.

State part (a)

Each side has $14$ tiles.

Set up the border count

Four sides of $14$ tiles each is a first estimate.

Multiply

That gives $56$.

Adjust for corners

The $4$ corner tiles were counted twice, so subtract $4$.

State part (b)

There are $52$ tiles on the border.

Sense check

Total tiles minus the $12$x1$2$ interior also gives $52$ border tiles.

Final answer

The requested final answer is the border count, $52$.

Set up the doubling

Doubling means multiply by $2$ every step.

Doubling steps $1$ to $3$

After three steps the value is $8$.

Doubling steps $4$ to $5$

After five steps the value is $32 = 2^5$.

Set up the tripling

Tripling means multiply by $3$ every step.

Tripling steps $1$ to $3$

After three steps the value is $27$.

Tripling steps $4$ to $5$

After five steps the value is $243 = 3^5$.

Compare the totals

The tripling result is larger.

Identify the larger

So $3^5$ is the larger.

Find the difference

Subtract to see by how much.

State the answer

$3^5$ beats $2^5$ by $211$.

Watch the gap grow

Early on the gap is tiny, then it widens fast.

Continue the gap

The gap balloons because tripling multiplies by more each time.

Reason about growth

A larger multiplier each step makes the gap widen quickly.

Sense check

$3^5$ is more than seven times $2^5$, so the large gap makes sense.

Underline the final answer

Write the answer on its own line so it is clear and easy to mark.

Understand the task

We need the number of $5$s that multiply to $625$.

First power

One $5$.

Second power

$5 \times 5 = 25$.

Third power

$5^3 = 125$.

Fourth power

$5^4 = 625$.

Match to the target

This equals the target $625$.

State k

So the power is $4$.

Verify with place value

Since $5^4 = (5^2)^2 = 25^2$, check $25$ x $25$.

Multiply the tens part

Break $25$ into $20 + 5$; first the $20$.

Multiply the units part

Then the $5$.

Add the parts

The parts add to $625$, confirming $25^2 = 625$.

Express another way

So $625$ is also the square of $25$.

Reject the near-miss

The common wrong answer $k = 3$ gives only $125$.

Conclude

$625$ is the $4$th power of $5$, so $k = 4$.

Underline the final answer

Write the answer on its own line so it is clear and easy to mark.

Spot the rule

The n-th square number is n squared.

List the first five squares

These are the first five terms given.

Continue the list

Carry on to the tenth term.

Count to the $10$th term

The tenth square uses $n = 10$.

Compute directly

$10 \times 10 = 100$.

State the answer

The $10$th square number is $100$.

Check another way: odd-number sums

Each square is the running total of the odd numbers.

Explain the pattern

The n-th square is the sum of the first n odd numbers.

Write the first ten odd numbers

These are the ten odds we must add.

Add in stages

The first five odds total $25$.

Add the next five

The next five odds total $75$.

Combine

The ten odd numbers total $100$.

Cross-check with the rule

Both methods give $100$.

Interpret as an array

$100$ is the number of dots in a $10$ by $10$ square array.

Conclude

So the $10$th square number is $100$.

Link area to side

Area of a square is side squared.

Set up the root

Take the square root of the area.

Test $100$

$100 \times 100 = 10000$.

State the side

Each side is $100$ m.

Recall perimeter

A square has four equal sides.

Compute the perimeter

The perimeter is $400$ m.

Multiply by the cost

Each metre costs $5$ pounds.

State the cost

The fencing costs $2000$ pounds.

Sense check

The side gives back the area, so the working is consistent.

Underline the final answer

Write the answer on its own line so it is clear and easy to mark.

Check it answers the question asked

Re-reading the question, £$2000$ is exactly the quantity required.

Sense-check the size

The size of £$2000$ is sensible for this calculation, so no slip has been made.

Confirm the answer is exact

Because the numbers are perfect powers or roots, £$2000$ is exact and needs no rounding.

Recap the method

The method was: apply the definition, work step by step, then check, giving £$2000$.

Check no factor was dropped

Counting the factors used matches the power, confirming £$2000$.