Math Vale

🔢 Number Kingdom · Number Theory

Square & Cube Roots

Undo a square or a cube: find the root of a perfect square or cube, solve x2 = p and x3 = p, and estimate the roots that are not whole numbers.

In short

  • A square root asks for two equal factors and a cube root asks for three: sqrt(49) = 7 because 7 x 7 = 49, and cbrt(64) = 4 because 4 x 4 x 4 = 64.
  • x2 = p has two solutions, one positive and one negative, while x3 = p has exactly one, whose sign matches p.
  • To estimate a root that is not whole, trap it between the perfect squares either side, then test tenths by squaring them.
  • Area gives the side of a square through a square root, and volume gives the edge of a cube through a cube root.
  • To compare a root with a whole number, square (or cube) the whole number instead of working the root out.

A root undoes a power

Squaring takes 7 to 49. A square root walks that journey backwards: sqrt(49) = 7, because 7 x 7 = 49.

Cubing takes 4 to 64. A cube root undoes that: cbrt(64) = 4, because 4 x 4 x 4 = 64.

That is the whole idea. Every root question is really the same question asked in reverse — *which number, multiplied by itself the right number of times, lands on this one?*

  • sqrt(p) asks for two equal factors: sqrt(81) = 9 because 9 x 9 = 81.
  • cbrt(p) asks for three equal factors: cbrt(27) = 3 because 3 x 3 x 3 = 27.

A number whose square root is whole is called a perfect square — 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 and onwards. A number whose cube root is whole is a perfect cube — 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000. Learning those two lists by sight is the single biggest time-saver in this topic, because most root questions are built from them.

Solving x^2 = p and x^3 = p

An equation like x2 = 49 is not quite the same as "what is sqrt(49)?", and the difference matters.

A negative number multiplied by itself gives a positive result: (-7) x (-7) = 49, exactly as 7 x 7 = 49. So x2 = 49 has two solutions, x = 7 and x = -7. People often write that as x = 7 or x = -7.

  • x2 = p (with p positive) always has two solutions, one positive and one negative.
  • The symbol sqrt(49) on its own means only the positive one, 7. That is a convention, not a contradiction — it keeps sqrt from meaning two things at once.
  • x2 = 0 has the single solution x = 0, and x2 = -25 has no solution among the numbers you know, because no number times itself is negative.

Cubes behave differently, and more simply. A cube keeps the sign of its base: 43 = 64 is positive, and (-4)3 = -64 is negative. So x3 = p has exactly one solution, and it carries the same sign as p. x3 = 125 gives x = 5; x3 = -125 gives x = -5.

If a coefficient is in the way, clear it first. For 3x2 = 108, divide both sides by 3 to get x2 = 36, and only then take roots: x = 6 or x = -6.

Estimating a root that is not whole

Most numbers are not perfect squares, and sqrt(53) is not a whole number. You can still pin it down closely with nothing but the times tables.

Step one: trap it between two whole numbers. Roots keep order — if one number is larger than another, so is its root. The perfect squares either side of 53 are 49 = 72 and 64 = 82, so

7 < sqrt(53) < 8

Step two: hunt for the first decimal place by squaring guesses. 7.52 = 56.25, which overshoots 53, so the answer is below 7.5. Try 7.22 = 51.84 (a little low) and 7.32 = 53.29 (just above). 53 sits between those, and it is nearer 53.29, so sqrt(53) is about 7.3.

Squaring your guess is the check that never lies. It turns "I think it is around 7.3" into something you can verify on paper.

The same trapping trick answers comparison questions without any estimating at all. Is sqrt(70) bigger than 8? Square the 8: 82 = 64, and 70 is bigger than 64, so sqrt(70) is bigger than 8. Comparing the numbers *inside* the roots is far quicker than working the roots out.

Where roots come from: area and volume

Squares and cubes are named after shapes, and the roots take you back to the shape's measurements.

A square of side s has area s x s = s2. So if a square room covers 196 m2, its wall length is sqrt(196) = 14 m. Notice the units: area is measured in m2 (square metres) and the root brings you back to plain m.

A cube of edge e has volume e x e x e = e3. So a cubical stone block of volume 343 cm3 has edges of cbrt(343) = 7 cm. Volume is in cm3, and the cube root brings you back to cm.

This is also why the shape names stick: 52 is read "5 squared" because it is the area of a square of side 5, and 53 is "5 cubed" because it is the volume of a cube of edge 5.

Roots turn up again the moment you meet the Pythagorean theorem, where the last step of every calculation is taking a square root of a total of squares.

Worked examples

Example 1

Solve 3x2 = 108.

  1. Get x2 on its own first: divide both sides by 3.
  2. x2 = 108 / 3 = 36.
  3. Now take square roots. 6 x 6 = 36, and (-6) x (-6) = 36 as well.
  4. So x = 6 or x = -6. Both answers count.

Example 2

A cubical crate has a volume of 512 cm3. How long is each edge?

  1. Volume of a cube = edge x edge x edge, so edge^3 = 512.
  2. That calls for a cube root, not a square root: edge = cbrt(512).
  3. Work up through the cubes: 73 = 343 (too small), 83 = 512 (exactly right).
  4. Each edge is 8 cm. Check: 8 x 8 x 8 = 512.

Example 3

Estimate sqrt(30) to one decimal place.

  1. Trap it first: 52 = 25 and 62 = 36, so sqrt(30) is between 5 and 6.
  2. Test the middle: 5.52 = 30.25, which is just above 30, so the answer is a little below 5.5.
  3. Test 5.4: 5.42 = 29.16, which is below 30. So the answer lies between 5.4 and 5.5.
  4. 30 is nearer 30.25 than 29.16, so sqrt(30) is about 5.5.

Practice problems, with solutions

Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.

Problem 1

Difficulty 1 of 5

sqrt(81) = ?

Answer: 9

  1. Look for n with n2 = 81.
  2. 9 x 9 = 81
  3. So sqrt(81) = 9.

Problem 2

Difficulty 3 of 5

Solve for x: x2 = 289 This equation has two solutions. Enter them as (a, b) with the smaller one first.

Answer: (-17, 17)

  1. x2 = 289
  2. 17 x 17 = 289, and (-17) x (-17) = 289.
  3. So x = 17 or x = -17, written (-17, 17).

Problem 3

Difficulty 4 of 5

Solve for x: x3 = 1,000 A cube has exactly one answer, so give that single value.

Answer: 10

  1. x3 = 1,000
  2. 10 x 10 x 10 = 1,000
  3. So x = cbrt(1,000) = 10.

Common mistakes

  • Halving instead of rooting: reading sqrt(36) as 18. A root shrinks a number far more than halving does.
  • Giving only the positive answer to x2 = 49. The negative root is a genuine solution, because (-7)2 = 49 too.
  • Expecting a plus-or-minus answer for a cube. x3 = 64 has just one solution, x = 4, because (-4)3 = -64, not 64.
  • Mixing the two roots up: taking sqrt(64) = 8 when the question asked for cbrt(64) = 4.
  • Stopping at x2 rather than x. Once x2 = 36 is on the page, one step is still left.
  • Adding roots as if sqrt(9 + 16) were 3 + 4. It is sqrt(25) = 5, not 7 — roots do not split across a sum.

What you should be able to do

  • Evaluate square roots of perfect squares and cube roots of perfect cubes.
  • Solve equations of the form x2 = p and x3 = p, remembering both square roots.
  • Estimate a non-perfect square root between two whole numbers and to one decimal place.
  • Use roots to find the side of a square or the edge of a cube from its area or volume.

Where this fits in the curriculum

Common Core

  • 8.EE.A.2

    Grade 8 — Use square root and cube root symbols to represent solutions to x^2 = p and x^3 = p, and evaluate roots of small perfect squares and cubes.

  • 8.NS.A.2

    Grade 8 — Use rational approximations of irrational numbers to compare them and locate them approximately on a number line.

Ontario

  • G8.B1.3

    Grade 8 — Estimate and calculate square roots, in various contexts.

  • G8.B2.2

    Grade 8 — Understand and recall commonly used square numbers and their square roots.

    Ontario names square roots only; cube roots are not a Grade 8 expectation there and ride in on the Common Core tag.

SAT

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