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Cubes and Cube Roots

Cube positive and negative numbers, fractions and decimals, find cube roots, and use them to solve volume problems and equations like x³ = −27.

Practice this skill 25 min Cubes and cube roots Leer en español
Key idea

What you'll learn

In this lesson you will learn to:

  • cube a number — whole numbers, negatives, fractions and decimals;
  • find the cube root of a number, including negative numbers;
  • go back and forth between the volume of a cube and the length of its edge;
  • solve equations like x^3 = -27 and 2x^3 + 10 = -44.

Why it matters on the GED: cubes appear in volume problems (boxes, tanks, ice blocks), in exponent questions, and in quick no-calculator items such as "What is \sqrt[3]{-64}?". The test likes to see whether you know the big difference between square roots and cube roots: a negative number has a real cube root (but no real square root).

In real life: a cube-shaped storage bin that holds 27 cubic feet has edges of 3 feet. Knowing how cubing works also explains why doubling the size of a box multiplies its volume by 8.

Words to know

  • Cube (of a number) — the number used as a factor three times: 4^3 = 4 \times 4 \times 4 = 64.
  • Exponent 3 / "cubed" — n^3 is read "n cubed" or "n to the third power".
  • Perfect cube — the cube of an integer: 1, 8, 27, 64, 125, \dots and also -1, -8, -27, \dots (-8 = (-2)^3).
  • Cube root — the number that, cubed, gives the original number: \sqrt[3]{64} = 4 because 4^3 = 64.
  • Index — the small number in the "notch" of the root sign: in \sqrt[3]{27} the index is 3 (cube root). A plain \sqrt{\phantom{x}} has index 2 (square root).
  • Cube (solid) — a box whose length, width and height are equal; each of these is an edge.
  • Volume — the space inside a solid, measured in cubic units (\text{ft}^3, \text{in}^3, \text{cm}^3): a cube with edge 2 in has volume 8\text{ in}^3.

All Math words in the glossary

1. Cubing numbers

To cube a number, multiply it by itself three times: n^3 = n \times n \times n.

n12345678910
n^31827641252163435127291000

Memorize the cubes from 1^3 to 10^3 (at least up to 5^3 = 125).

A cube with edges 3 units long, built from 27 small unit cubes (3 by 3 by 3), showing that 3 cubed equals 27.333Volume = 3³ = 27 cubic units

The picture shows a cube with edges of 3 units. It holds 3 \times 3 \times 3 = 27 small unit cubes, so its volume is 3^3 = 27 cubic units.

Cubing is not tripling. 4^3 = 64, not 4 \times 3 = 12.

Negatives keep their sign. Multiply step by step: (-2)^3 = (-2)(-2)(-2) = (4)(-2) = -8. Three negative factors → negative. So:

  • the cube of a positive number is positive;
  • the cube of a negative number is negative;
  • 0^3 = 0.

(Compare with squares: (-2)^2 = +4.) As with squares, parentheses matter: -2^3 = -(2^3) = -8, and here (-2)^3 happens to also equal -8.

Fractions: cube the top and the bottom. \left(\tfrac{2}{3}\right)^3 = \tfrac{8}{27} and \left(-\tfrac{1}{3}\right)^3 = -\tfrac{1}{27}.

Decimals: cube the digits, then triple the number of decimal places. 0.2^3 = 0.008 (three places), 1.5^3 = 3.375, 0.1^3 = 0.001.

2. Cube roots

The cube root undoes cubing. \sqrt[3]{125} asks: what number, used three times as a factor, gives 125? Answer: 5.

  • \sqrt[3]{8} = 2, \sqrt[3]{27} = 3, \sqrt[3]{1000} = 10, \sqrt[3]{1} = 1, \sqrt[3]{0} = 0.

Cube roots of negative numbers are real. \sqrt[3]{-64} = -4, because (-4)^3 = -64. This is the big difference from square roots: \sqrt{-64} is not a real number, but \sqrt[3]{-64} = -4 is.

Every number has exactly one real cube root, and it has the same sign as the number. So x^3 = 27 has only one real solution, x = 3 (not \pm 3, since (-3)^3 = -27).

Graph of y = x³ for x from about −2 to 2, passing through (−2, −8), (−1, −1), (0, 0), (1, 1) and (2, 8). A dashed red line at y = −8 meets the curve only once, at x = −2, so ∛(−8) = −2; a dashed green line at y = 8 meets it once at x = 2, so ∛8 = 2. Negative numbers have negative cube roots.y = x³: every number has exactly one real cube root−8−448xy = x³−2−112(2, 8)(−2, −8)∛(−8) = −2∛8 = 2

Fractions and decimals:

\sqrt[3]{\frac{8}{125}} = \frac{\sqrt[3]{8}}{\sqrt[3]{125}} = \frac{2}{5} \qquad \sqrt[3]{0.001} = 0.1 \qquad \sqrt[3]{0.027} = 0.3

(The cube root has one third as many decimal places.)

Products split; sums do not. \sqrt[3]{8 \cdot 27} = 2 \cdot 3 = 6 ✓, but \sqrt[3]{8} + \sqrt[3]{27} = 5 while \sqrt[3]{8 + 27} = \sqrt[3]{35} \approx 3.27 ✗.

Estimating: \sqrt[3]{100} is between \sqrt[3]{64} = 4 and \sqrt[3]{125} = 5 (calculator: \approx 4.64).

3. Cubes in volume problems and equations

Volume of a cube. A cube with edge length s has volume V = s^3. Working backwards, if you know the volume, the edge is s = \sqrt[3]{V}.

  • Edge 5 in → V = 5^3 = 125\text{ in}^3.
  • Volume 343\text{ ft}^3 → edge \sqrt[3]{343} = 7\text{ ft}.

Scaling. If every edge of a cube is multiplied by k, the volume is multiplied by k^3. Doubling the edges (k = 2) multiplies the volume by 2^3 = 8, not by 2. Example: a 3-ft cube holds 27\text{ ft}^3; a 6-ft cube holds 216\text{ ft}^3 = 8 \times 27.

Left: a cube with 3-foot edges, volume 3³ = 27 cubic feet. Right: a cube with 6-foot edges, split into 2 by 2 by 2 = 8 copies of the 3-foot cube. Its volume is 6³ = 216 cubic feet, which is 8 × 27: doubling the edge multiplies the volume by 8.Double the edge → volume × 2³ = 83 ft3³ = 27 ft³3 ft3 ft6 ft6³ = 216 ft³ = 8 × 27

Solving equations with a cube. Isolate x^3 first (undo adding and multiplying), then take the cube root:

2x^3 + 10 = -44 \;\Rightarrow\; 2x^3 = -54 \;\Rightarrow\; x^3 = -27 \;\Rightarrow\; x = \sqrt[3]{-27} = -3

Don't confuse the three measurements of a cube with edge s:

MeasurementFormulaEdge 4
Area of one faces^216
Surface area (6 faces)6s^296
Volumes^364
Left: a cube with edge s = 4 built from 4 × 4 × 4 = 64 small cubes; its front face is outlined in red with area 4² = 16, and its volume is 4³ = 64. Right: the cube unfolded into a net of 6 square faces, each with area 16, so the surface area is 6 × 16 = 96.Cube with edge s = 4: face, surface area, volumes = 4face4² = 16volume = 4³ = 64 (4 × 4 × 4 small cubes)161616161616surface area = 6 faces6 × 16 = 96
Formula

Formulas and rules

n^3 = n \cdot n \cdot n \qquad \left(\sqrt[3]{n}\right)^3 = n \qquad \sqrt[3]{n^3} = n \sqrt[3]{-n} = -\sqrt[3]{n} \qquad \sqrt[3]{ab} = \sqrt[3]{a}\,\sqrt[3]{b} \qquad \sqrt[3]{\frac{a}{b}} = \frac{\sqrt[3]{a}}{\sqrt[3]{b}}

Volume of a cube: V = s^3, where s is the edge length; so s = \sqrt[3]{V}.

Formula sheet: the GED formula sheet gives the volume of a rectangular prism, V = lwh (and of any right prism, V = Bh, base area × height). For a cube, the length, width and height are all s, so V = s \cdot s \cdot s = s^3. The cube shortcut V = s^3 and all the root rules above are not printed on the sheet — memorize them, along with the perfect cubes up to 10^3 = 1000.

Worked example

Example 1 (easy): cube a negative

Evaluate (-3)^3.

Number line from −30 to 30. A dot at 9 shows (−3)(−3) = 9; a dashed red arrow labelled times −3 goes from 9 to −27. Dots mark (−3)³ = −27 in red and 3³ = 27 in blue: three negative factors give a negative cube.(−3)³ = (−3)(−3)(−3) = 9 × (−3) = −27−27−18−9091827(−3)³ = −273³ = 27(−3)(−3) = 9× (−3)Three negative factors (an odd number) → negative
  1. Write it out: (-3)(-3)(-3).
  2. First two factors: (-3)(-3) = 9.
  3. Times the third: 9 \times (-3) = -27.

Answer: -27

Check: An odd number (3) of negative factors gives a negative result, and 3^3 = 27 ✓.

Worked example

Example 2: cube root of a negative fraction

Evaluate \sqrt[3]{-\tfrac{125}{8}}.

Number line from −3 to 3 in halves with a blue dot at −2.5, labelled −5/2. Check: (−5/2)³ = −125/8 = −15.625, so the cube root of −125/8 is −5/2.∛(−125/8) = −∛125 / ∛8 = −5/2 = −2.5−3−2−10123−5/2 = −2.5check: (−5/2)³ = −125/8 = −15.625Negative number in → negative cube root out
  1. The number is negative, so the cube root is negative.
  2. Root of the top: \sqrt[3]{125} = 5. Root of the bottom: \sqrt[3]{8} = 2.
  3. So the value is -\tfrac{5}{2} = -2\tfrac{1}{2} = -2.5.

Answer: -\tfrac{5}{2}

Check: \left(-\tfrac{5}{2}\right)^3 = -\tfrac{125}{8} ✓.

Worked example

Example 3 (word problem, work backwards): edge from volume

A cube-shaped storage bin holds 64 cubic feet. How much floor space (the area of its bottom face) does it take up?

Left: a cube-shaped bin with 4-foot edges, built from 4 × 4 × 4 = 64 one-foot cubes, so its volume is 64 cubic feet. Right: its bottom face, a 4 ft by 4 ft square of 16 one-foot squares, so the floor space is 4² = 16 square feet.V = s³ = 64 ft³ → s = ∛64 = 4 ft4 ft4 ftbottom face (floor space)4 ft4 ft4² = 16 ft²
  1. Volume of a cube: s^3 = 64, so s = \sqrt[3]{64} = 4 ft.
  2. The bottom face is a 4\text{ ft} \times 4\text{ ft} square: 4^2 = 16.

Answer: 16 square feet

Check: 4 \times 4 \times 4 = 64\text{ ft}^3 ✓, and the bottom face is 16\text{ ft}^2 ✓.

Worked example

Example 4 (word problem): cube a decimal

A concrete block is a cube with edges of 1.2 meters. What is its volume?

A concrete cube with edges labelled 1.2 m and volume 1.728 cubic meters, drawn next to a 1 m cube (1 cubic meter) at the same scale. 1.2 × 1.2 × 1.2 = 1.728, so the block holds about 1.7 times as much as the 1 m cube.V = 1.2 × 1.2 × 1.2 = 1.728 m³1.2 m1.2 m1.2 m1 m1 m³1.728 m³Same scale: a 1-m cube holds 1 m³, so 1.728 m³ is about 1.7 times as much.
  1. V = s^3 = 1.2^3.
  2. 1.2 \times 1.2 = 1.44.
  3. 1.44 \times 1.2 = 1.728.

Answer: 1.728 cubic meters

Check: Estimate: 1^3 = 1 and 1.5^3 \approx 3.4, so about 1.7 is reasonable. Decimal places: 1.2 has 1, cubed gives 3 places ✓.

Worked example

Example 5 (GED-hard): solve an equation

Solve 3x^3 - 5 = 19.

Step diagram solving 3x³ − 5 = 19: add 5 to both sides to get 3x³ = 24; divide both sides by 3 to get x³ = 8; take the cube root to get x = 2.3x³ − 5 = 19add 5 to both sides3x³ = 24divide both sides by 3x³ = 8take the cube root: ∛8 = 2x = 2
  1. Add 5 to both sides: 3x^3 = 24.
  2. Divide both sides by 3: x^3 = 8.
  3. Take the cube root: x = \sqrt[3]{8} = 2.

Answer: x = 2

Check: 3(2)^3 - 5 = 3(8) - 5 = 24 - 5 = 19 ✓. (Unlike x^2 = 8, there is only one real solution; (-2)^3 = -8, not 8.)

Calculator tip

Using the TI-30XS MultiView

  • Cube a number: type the base, press ^, type 3, press enter. For a negative, use parentheses and the (−) key: ( (−) 3 ) ^ 3 → -27. For a fraction base, put it in parentheses or use the n/d template first.
  • Cube root: type the index 3 first, then press 2nd ^ to get the root \sqrt[x]{\phantom{x}}, type the number (for example (−) 64), then enter → -4.
  • Decimal ↔ fraction: press toggle to switch, for instance between \tfrac{8}{27} and 0.296\ldots.
  • Check by cubing back: after finding a cube root, cube the result to see if you get the original number.

When NOT to use it: cubes and cube roots of small perfect cubes (8, 27, 64, 125, 1000 and their negatives) are fair game in the no-calculator section. Know them by heart; save the calculator for decimals like 1.2^3 or \sqrt[3]{100}.

Common mistake

Common mistakes

  • Tripling instead of cubing. Wrong: 4^3 = 12. Right: 4 \times 4 \times 4 = 64.
  • Squaring instead of cubing. Wrong: 5^3 = 25. Right: 5^3 = 125. Count the factors: the exponent 3 means three of them.
  • Thinking a cube of a negative is positive. Wrong: (-2)^3 = 8. Right: (-2)(-2)(-2) = -8. An odd number of negatives is negative.
  • Saying the cube root of a negative is undefined. Wrong: "\sqrt[3]{-27} does not exist". Right: \sqrt[3]{-27} = -3. Only even roots (like square roots) of negatives are not real.
  • Dividing by 3 instead of taking the cube root. Wrong: \sqrt[3]{27} = 9. Right: \sqrt[3]{27} = 3.
  • Doubling the volume when the edge doubles. Wrong: edge × 2 → volume × 2. Right: volume × 2^3 = 8.
Test strategy

GED test strategy

  • Sign first. For any cube or cube root, the answer has the same sign as the number you start with. Cross out choices with the wrong sign immediately.
  • Cube the answer choices. In "what is \sqrt[3]{\ldots}" or "find the edge" questions, cube each choice; only one will give the original number.
  • Watch for "undefined" as a trap choice. It is correct for even roots of negatives, never for cube roots.
  • Know which measurement is asked. Face area (s^2), surface area (6s^2) and volume (s^3) of the same cube are three different numbers; all three are likely answer choices.
  • Units: volume is in cubic units; an edge found from a volume is in plain units.
  • Estimate with perfect cubes (4^3 = 64, 5^3 = 125) to check a calculator result in seconds.

Check yourself

Cubes and cube rootsNo calculator

What is the value of 4^3?

Answer choices
Hint

The exponent 3 means "use 4 as a factor three times".

Check yourself

Cubes and cube rootsNo calculatorFill in

What is the value of \sqrt[3]{125}?

Type your answer in the box. Fractions like 3/4 are fine.

Hint

Which whole number, multiplied by itself three times, equals 125?

Summary

Recap

  • n^3 = n \cdot n \cdot n; memorize 1^3 to 10^3 (1, 8, 27, 64, 125, 216, 343, 512, 729, 1000).
  • Cubing keeps the sign: (-2)^3 = -8. Fractions: cube top and bottom. Decimals: triple the decimal places.
  • \sqrt[3]{\phantom{x}} undoes cubing; negative numbers have real cube roots: \sqrt[3]{-64} = -4.
  • Cube of edge s: V = s^3 and s = \sqrt[3]{V}. Doubling the edge multiplies the volume by 8.
  • Solve ax^3 + b = c by isolating x^3, then taking the cube root — there is only one real solution.
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