Square, Square Root, Cube and Cube Root

The definition of a square and the squares from 1 to 30.

Multiplying a number by itself gives its square. $6^2=6\times 6=36$. A number whose square root is a whole number is a perfect square — 36, 49 and 144 are; 40 is not. Learn the squares from 1 to 30 by heart; every shortcut later rests on them.

By formula, numbers ending in 5, numbers near 100, and decimals.

1) By the formula

2) A number ending in 5

$(10k+5{)}^2=100k(k+1)+25$. Multiply the number before the 5, $k$, by the next number $k+1$, and put 25 after it. $75^2$: $7\times 8=56$ → 5625. $125^2$: $12\times 13=156$ → 15625.

3) A number near 100

$(100+d{)}^2=10000+200d+d^2$. For $104^2$: $d=4$ → $10000+800+16=10816$. For $96^2$: $d=-4$ → $10000-800+16=9216$. Add or subtract twice the difference from 100, then add the square of the difference.

4) Squaring a decimal

Ignore the point, square the number, then put twice as many decimal places in the answer. $3.5^2$: $35^2=1225$; one place in the number, so two in the answer → 12.25. $0.04^2$: $4^2=16$, 2 places × 2 = 4 places → 0.0016.

Checks that eliminate options without solving.

• No perfect square ends in 2, 3, 7 or 8.
• A perfect square never ends in an odd number of zeros (like 4000).
• Squares of consecutive numbers differ by the sum of the two numbers: $(n+1{)}^2-n^2=2n+1$. E.g. $12^2-11^2=23$.
• The sum of the first $n$ odd numbers is $n^2$: $1+3+5+7+9=25=5^2$.
• A perfect square with $n$ digits has a root with $\frac{n}{2}$ digits if $n$ is even, $\frac{n+1}{2}$ if $n$ is odd.

The cleanest way for small and medium perfect squares.

When factors are hard: pair the digits, bring down, double the divisor.

Pairs are made on both sides of the point; for a fraction take the root of the top and bottom separately.

The favourite question of the exams, in four forms.

The definition, the cubes from 1 to 20, and cubing a larger number by formula.

Two methods; the second works in seconds on multiple-choice questions.

Digits in the root, negative cubes and the questions that make a perfect cube.

The slips that cost marks.

Remove half the options without solving.

One-rule questions.

Exam-level questions.

Questions at the hard end of competitive exams.