EQN-01-06

Squares, Roots & Scientific Notation

Recognize perfect squares and cubes, simplify square roots, and read scientific notation at a glance.

~5 min read•Difficulty 2/5

Know these by heart

Recognizing these on sight turns a root into instant recall.

Perfect squares (1² to 15²):

n	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
n²	1	4	9	16	25	36	49	64	81	100	121	144	169	196	225

Perfect cubes (1³ to 5³):

n	1	2	3	4	5
n³	1	8	27	64	125

Square roots

$n$ asks "what squared gives n?" $. 49 = 7.$ To simplify a non-perfect root, pull out the largest perfect-square factor:

50 = 25 \cdot 2 = 52

Roots distribute over products and quotients, but not over sums $: 9 + 16 = 25 = 5,$ which is NOT $3 + 4.$

Scientific notation

A number written as $a \times 1 0^{n}$ with $1 \leq a < 10.$

3200 = 3.2 \times 1 0^{3} 0.0047 = 4.7 \times 10 ⁻^{3}

Positive exponent $\to$ large number (move the point right); negative $\to$ small number (move left). To multiply, multiply the fronts and add the exponents.

Why it matters on the GAT-E

It lets you compare or estimate huge/tiny numbers without writing all the zeros — perfect for comparison questions.

Worked examples

Example 1 (difficulty 2/5)

Simplify

72.

◎ Predict the answer first:

Example 2 (difficulty 2/5)

Write 0.00056 in scientific notation.

◎ Predict the answer first:

Tips

Memorize squares to 15² and cubes to 5³ — they appear in geometry and algebra constantly.
To simplify a root, factor out the largest perfect square, not just any factor.
Roots and exponents distribute over $\times$ and $\div$ but never over $+$ or $- .$

Common mistakes

Splitting a root over a sum

: a + b = a + b

Why: It mirrors how multiplication distributes.

Fix: Roots distribute only over products/quotients; add under the root first.

Getting the sign of a scientific-notation exponent backwards

Why: Small numbers feel like they should have a positive power.

Fix: Numbers below 1 use a negative exponent; numbers above 10 use a positive one.

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