Squares, Roots, and Powers

The second page of the times table: the numbers the GED asks for by name.

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A square is a number times itself. A root reads the square backward. Once the twenty squares below are familiar, most of what the test calls exponents and roots is already in your hands. The rest of this page is the short list of powers, cubes, and right-triangle numbers that turn up again and again.

How this works
  • Look. Tap any square to see it both ways: forward as a multiplication, backward as a root. Hide the answers to test yourself.
  • Practice. Questions on a keypad, mixed from everything on this page. Each right answer polishes a fact. After three, it shines. A wrong answer takes one polish away and brings the fact back soon. Your progress stays on this device.
Tap a square for the whole story.

The twenty squares, and their roots

The first twelve are on the diagonal of the times table. 13 to 16 are the ones the test expects you to know on sight: 169, 196, 225, 256. And 25² = 625 turns up in area problems often enough to earn a place.

Round numbers, and powers of ten

The rule, not the list: square the digit, then write the zeros twice. 30² is 3² with two zeros twice: 900. 1,000² is 1 with three zeros twice: 1,000,000.

This is why scientific notation works. 3.2 × 10⁴ means 3.2 with the point moved four places: 32,000. And 0.0045 is 4.5 × 10⁻³, the point moved three places the other way.

Power chains

Each row multiplies by the same number again and again. The exponent counts how many times.

Cubes are the third column, read down: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1,000. A cube root reads them backward: ∛64 = 4, ∛125 = 5. On the test that is the edge of a cube whose volume you know.

Three rules you can check on the chain. Anything to the 1 is itself: 7¹ = 7. Anything to the 0 is 1: 7⁰ = 1. And when you multiply two powers of the same number, the exponents add: 2⁴ × 2² = 2⁶ — look at the chain, 16 × 4 = 64. A negative exponent flips the number over: 2⁻¹ = ½, 10⁻² = 1/100.

Roots that are not whole numbers

The test asks it this way: √50 is between which two whole numbers?

Look at the squares. 49 is 7². 64 is 8². 50 sits between them, so √50 is between 7 and 8 — and close to 7, since 50 is only one past 49. No calculator needed; the table of squares is the whole method.

Try these, then open each one.

√20 is between ? and ?

16 is 4² and 25 is 5². So √20 is between 4 and 5, a little closer to 4.

√90 is between ? and ?

81 is 9² and 100 is 10². So √90 is between 9 and 10, about halfway.

√150 is between ? and ?

144 is 12² and 169 is 13². So √150 is between 12 and 13, closer to 12.

Right-triangle numbers

4 3 5

In a right triangle, the two short sides squared and added give the long side squared: a² + b² = c².

3² + 4² = 9 + 16 = 25, and 25 is 5². So the long side is 5. The squares table does the work; the formula only tells you which squares to add.

The families the test uses. Nearly every right-triangle question on the GED is one of these, or one of these with every side doubled or tripled.

A ladder against a wall, a ramp, the diagonal of a rectangle, the distance between two points on a grid — all the same question. If two sides are on this list, the third is too.