Square roots & cube roots — notes, question types & practice · Simplification

01

Overview

Every root question in this chapter is built on a perfect power, and the exam expects you to find it without long division. Two tools do it: the unit digit of the answer, and the size of the answer from digit grouping.

02

The unit-digit maps

A perfect square ends only in $0, 1, 4, 5, 6$ or $9$ ; the digits $2, 3, 7, 8$ never appear. The map from root to square is short: root ending $1$ or $9$ gives square ending $1$ . Root ending $2$ or $8$ gives $4$ , and $3$ or $7$ gives $9$ . Root ending $4$ or $6$ gives $6$ , and $5$ gives $5$ . Cube digits are even friendlier: $1, 4, 5, 6, 9, 0$ map to themselves, and $2$ swaps with $8$ , $3$ with $7$ .

Rule: For a cube root, the last digit of the cube gives the last digit of the root directly, using the swap pairs 2 with 8 and 3 with 7.

03

Size from digit grouping

Pair the digits of a square from the right: the number of pairs equals the digits in the root. Group a cube in threes the same way. For $\sqrt{9604}$ : pairs $96 \mid 04$ , so a two-digit root; $9^2 = 81 \le 96$ , so it starts with $9$ . Ending $4$ means root ends in $2$ or $8$ ; since $95^2 = 9025$ is too small, the root is $98$ . For a cube root, the leading group fixes the first digit: the largest cube not exceeding it. In $\sqrt[3]{175616}$ , groups are $175 \mid 616$ ; $5^3 = 125 \le 175 < 216 = 6^3$ , so the root starts with $5$ and ends with $6$ : answer $56$ .

04

Roots of decimals

Count decimal places. A square root halves them: $0.0169$ has four places, so its root has two: $0.13$ . A cube root divides them by three: $0.000064$ has six places, root $0.04$ . Combine both: $\sqrt{1.69} + \sqrt{0.0169} = 1.3 + 0.13 = 1.43$ .

Watch: A unit digit can only rule a perfect square out, never confirm it. Always finish with the size check.

05

Least number to add or subtract

Bracket $N$ between neighbouring squares $k^2$ and $(k+1)^2$ .

Subtract case: answer is $N - k^2$ , the gap down.
Add case: answer is $(k+1)^2 - N$ , the gap up. For $456$ : $21^2 = 441$ and $22^2 = 484$ , so subtracting $456 - 441 = 15$ lands on $441$ . For $2000$ : $44^2 = 1936$ and $45^2 = 2025$ , so adding $2025 - 2000 = 25$ lands on $2025$ .

06

Nested radicals: inside out

Replace the deepest root first; each layer then becomes a perfect square. $\sqrt{20 + \sqrt{19 + \sqrt{36}}}$ : innermost $\sqrt{36} = 6$ ; then $\sqrt{19 + 6} = \sqrt{25} = 5$ ; then $\sqrt{20 + 5} = \sqrt{25} = 5$ .

Tip: Setters place each layer to collapse exactly. If a layer does not land on a perfect square, recheck the inner layer before forcing on.

07

Infinite radicals

Call the whole expression $x$ ; the tail equals $x$ again, so $x = \sqrt{a + x}$ . Then $x^2 - x - a = 0$ ; keep the positive root. Fast path: if $a = k(k+1)$ , the answer is $k + 1$ . Check with $12 = 3 \times 4$ : $4^2 = 12 + 4$ holds, so $\sqrt{12 + \sqrt{12 + \cdots}} = 4$ . The product form $\sqrt{x\sqrt{x\sqrt{x \cdots}}}$ equals $x$ itself.

08

Root equations with an unknown

Isolate the root on one side and square once. $\sqrt{1 + \dfrac{x}{225}} = \dfrac{16}{15}$ gives $1 + \dfrac{x}{225} = \dfrac{256}{225}$ , so $x = 31$ . Square both sides only after the root stands alone.

09

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common3 practice Q

Square root or cube root of a perfect power

How to spot it:

A clean big number sits under a root sign and the options are whole numbers.

\text{digit pairs (squares) or triples (cubes) plus the unit-digit map}

Method

Group digits from the right: pairs for a square root, triples for a cube root.
Read the size from the leading group.
Read the last digit from the map or swap map.
Verify by squaring or cubing the candidate.

Why it works:

The grouping gives size, the unit digit gives the tail; together they pin the answer without long division.

Try this

Find $\sqrt[3]{175616}$ .

Show solution

Groups: $175 \mid 616$ , a two-digit root.
$5^3 = 125 \le 175 < 216 = 6^3$ : first digit $5$ .
Ending $6$ maps to itself: root $56$ ; check $56^3 = 175616$ .

Answer

56

Type 2very common2 practice Q

Least number to add or subtract to reach a perfect square

How to spot it:

The question asks for the least number added to or subtracted from N so the result is a perfect square.

\text{subtract: } N - k^2,\quad \text{add: } (k+1)^2 - N

Method

Find $k$ , the whole part of the square root of $N$ .
Compute $k^2$ and $(k+1)^2$ to bracket $N$ .
Subtract case: answer $N - k^2$ ; add case: answer $(k+1)^2 - N$ .

Why it works:

The nearest squares on the two sides of $N$ are exactly $k^2$ and $(k+1)^2$ .

Try this

Find the least number that must be added to $2000$ to make it a perfect square.

Show solution

$44^2 = 1936 < 2000 < 2025 = 45^2$ .
Gap up: $2025 - 2000 = 25$ .

Answer

25

Type 3common2 practice Q

Nested radicals, worked from the inside

How to spot it:

A finite stack of square roots, each wrapped around the previous result.

\text{evaluate the innermost root first}

Method

Replace the deepest root with its value.
Add or combine outward one layer at a time.
Each layer should land on a perfect square.

Why it works:

Only the innermost layer is a number you can root immediately; every outer layer waits for it.

Try this

Find $\sqrt{20 + \sqrt{19 + \sqrt{36}}}$ .

Show solution

$\sqrt{36} = 6$ .
$\sqrt{19 + 6} = \sqrt{25} = 5$ .
$\sqrt{20 + 5} = \sqrt{25} = 5$ .

Answer

5

Type 4common2 practice Q

Infinite radicals

How to spot it:

The same pattern repeats forever, shown by dots; the word infinite may appear.

x = \sqrt{a + x} \Rightarrow x^2 - x - a = 0

Method

Name the whole expression $x$ .
Spot the tail equals $x$ again and build the equation.
Solve and keep the positive root.
Fast path: factor $a$ as $k(k+1)$ ; the answer is $k + 1$ .

Why it works:

Self-similarity turns the endless tower into one short quadratic.

Try this

Find $\sqrt{12 + \sqrt{12 + \sqrt{12 + \cdots}}}$ .

Show solution

$x^2 = 12 + x$ , so $x^2 - x - 12 = 0$ .
$12 = 3 \times 4$ , so the answer should be $4$ .
Check: $4^2 = 16 = 12 + 4$ . So $x = 4$ .

Answer

4

Type 5common

Roots of decimals

How to spot it:

A decimal sits under the root sign, like $1.69$ or $0.000064$ .

\text{decimal places halve (square root) or divide by three (cube root)}

Method

Count the decimal places of the number.
Halve them for a square root; take a third for a cube root.
Root the digits as a whole number and place the point.

Why it works:

Rooting affects the place values exactly as the index divides the decimal count.

Try this

Find $\sqrt{1.69} + \sqrt{0.0169}$ .

Show solution

$\sqrt{1.69} = 1.3$ (two places become one).
$\sqrt{0.0169} = 0.13$ (four places become two).
$1.3 + 0.13 = 1.43$ .

Answer

1.43

10

Formula sheet

Product rule

\sqrt{ab} = \sqrt{a}\,\sqrt{b},\quad \sqrt[3]{ab} = \sqrt[3]{a}\,\sqrt[3]{b}

Infinite radical, plus

\sqrt{n + \sqrt{n + \cdots}} = \dfrac{1 + \sqrt{1 + 4n}}{2}

Infinite radical, minus

\sqrt{n - \sqrt{n - \cdots}} = \dfrac{-1 + \sqrt{1 + 4n}}{2}

Infinite nested product

\sqrt{x\sqrt{x\sqrt{x \cdots}}} = x

Least add or subtract

\text{subtract } N - k^2,\quad \text{add } (k+1)^2 - N

11

Shortcuts that save time

⚡ Cube root by unit digit and leading group

The last digit of the cube gives the last digit of the root; the leading group, compared with the small cubes, gives the first digit.

Example

Find the cube root of 175616.

Show solution

Groups: $175 \mid 616$ , so the root has two digits.
$5^3 = 125 \le 175 < 216 = 6^3$ , so first digit is $5$ .
Cube ends in $6$ , and $6$ maps to itself: root $= 56$ .

Answer

56

⚡ Square root by unit digit and size

Digit pairs fix the number of digits and the first digit; the last-digit map gives two candidates; one size check picks between them.

Example

Find the square root of 9604.

Show solution

Pairs $96 \mid 04$ : two digits; $9^2 = 81 \le 96$ , so it starts with $9$ .
Ends in $4$ , so the root ends in $2$ or $8$ .
$95^2 = 9025$ is below $9604$ , so take $98$ ; check $98^2 = 9604$ .

Answer

98

⚡ n equals k times k plus one radicals

Factor n into two consecutive integers. Plus signs give the larger one, minus signs the smaller.

Example

Find $\sqrt{72 + \sqrt{72 + \sqrt{72 + \cdots}}}$ .

Show solution

$72 = 8 \times 9$ , two consecutive integers.
Plus form gives the larger: $9$ .
Check: $9^2 = 81 = 72 + 9$ .

Answer

9

12

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Miscounting decimal places in roots of decimals.

Square root halves the decimal places; cube root divides them by three.

Mistake 02

Keeping the negative root when solving x squared equals a plus x.

A radical expression is positive; take the positive root of the quadratic.

Mistake 03

Declaring a number a perfect square because it ends in 1, 4, 5, 6 or 9.

The unit digit only rules candidates out; confirm with digit pairing and a check.

Mistake 04

Swapping the add and subtract gaps.

Subtract down to k squared; add up to the next square, k plus one squared.

Mistake 05

Opening a nested radical from the outermost root.

Evaluate from the innermost root outward, one layer at a time.

13

Quick revision

Read this the night before the exam.

Perfect squares end in $0, 1, 4, 5, 6, 9$ ; cubes use the swap map $2 \leftrightarrow 8$ , $3 \leftrightarrow 7$ .
Digit pairs size a square root; digit triples size a cube root.
Subtract $N - k^2$ ; add $(k+1)^2 - N$ .
Nested radicals open from the inside out.
$x = \sqrt{a + x}$ ; if $a = k(k+1)$ , the answer is $k + 1$ .
Product form $\sqrt{x\sqrt{x\cdots}}$ equals $x$ .

14

Practice: 13 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 13 questions

Suggested time 8 min · wrong answers go to your mistake notebook automatically.