Cubes and cube roots

1. 4^3 means 4 \times 4 \times 4.
2. 4 \times 4 = 16.
3. 16 \times 4 = 64.

Answer: 64

Check: Work backwards: 64 \div 4 = 16 and 16 \div 4 = 4, so 64 really is three 4s multiplied together ✓. On the TI-30XS, 4 ^ 3 enter also shows 64.

Tip: The exponent tells you how many times the base is multiplied by itself — it is never "base times exponent". Memorize the first few cubes: 1, 8, 27, 64, 125.

1. (-2)^3 = (-2) \times (-2) \times (-2).
2. The first two factors: (-2) \times (-2) = 4 (negative × negative = positive).
3. Times the third factor: 4 \times (-2) = -8 (positive × negative = negative).

Answer: -8

Check: Three negative factors is an odd number of negatives, so the product must be negative; and 2^3 = 8 ✓.

Tip: A negative number raised to an odd power stays negative; raised to an even power it becomes positive. (-2)^2 = 4 but (-2)^3 = -8.

1. 6^3 means 6 \times 6 \times 6 — three factors of 6.
2. 6 \times 6 = 36.
3. 36 \times 6 = 216.

Answer: 216

Check: 216 \div 6 = 36 and 36 \div 6 = 6, so 216 is made of three 6s multiplied together ✓. Estimate: 5^3 = 125 and 7^3 = 343, and 216 sits between them ✓.

Tip: Learn the first ten cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1,000. They appear in volume problems and cube-root questions, often on the no-calculator part.

1. Volume of a cube: V = s^3, so s = \sqrt[3]{V} = \sqrt[3]{1{,}728}.
2. Bracket with known cubes: 10^3 = 1{,}000 and 20^3 = 8{,}000, so the edge is between 10 and 20 — close to 10.
3. The last digit of 1,728 is 8, and 2^3 = 8, so try 12: 12 \times 12 = 144, 144 \times 12 = 1{,}728 ✓.

Answer: 12 inches

Check: 12^3 = 1{,}728 ✓. (Fun fact: 1,728 cubic inches is exactly 1 cubic foot, since 1 foot = 12 inches.)

Tip: The last digit of a perfect cube tells you the last digit of its root: cubes ending in 8 come from numbers ending in 2, ending in 7 from 3, ending in 4 from 4, ending in 5 from 5, and so on.

1. Volume of a cube: V = s^3.
2. V = 18^3 = 18 \times 18 \times 18.
3. 18 \times 18 = 324.
4. 324 \times 18 = 324 \times 20 - 324 \times 2 = 6{,}480 - 648 = 5{,}832.

Answer: 5,832 cubic inches

Check: Estimate: 20^3 = 8{,}000 and 15^3 = 3{,}375, so 18^3 should be between them, closer to 8,000 ✓.

Tip: Watch the units: an edge in inches gives a volume in cubic inches. 324 square inches is the area of one face — that is a common distractor.

1. Exponents come before addition. Cube each number separately.
2. 3^3 = 3 \times 3 \times 3 = 27.
3. 2^3 = 2 \times 2 \times 2 = 8.
4. Add: 27 + 8 = 35.

Answer: 35

Check: If you added first you would get (3 + 2)^3 = 5^3 = 125, which is much bigger than 35 — and it ignores the order of operations ✓.

Tip: a^3 + b^3 is not (a + b)^3. Always evaluate each power on its own before adding.

1. Volume of a cube: V = s^3, so s^3 = 343.
2. Undo the cube with a cube root: s = \sqrt[3]{343}.
3. Try perfect cubes: 6^3 = 216, 7^3 = 7 \times 7 \times 7 = 49 \times 7 = 343 ✓.
4. So s = 7 inches.

Answer: 7 inches

Check: 7 \times 7 \times 7 = 343\text{ in}^3 ✓. On the TI-30XS: 3 2nd ^ 343 enter → 7.

Tip: Volume → edge means cube root; area → side means square root. Using the wrong root (here \sqrt{343} \approx 18.5) is a classic trap.

1. \sqrt[3]{-64}: which number cubed is -64? Since 4^3 = 64, we get (-4)^3 = -64. So \sqrt[3]{-64} = -4.
2. \sqrt[3]{27} = 3, because 3^3 = 27.
3. Add: -4 + 3 = -1.

Answer: -1

Check: (-4)^3 = (-4)(-4)(-4) = 16 \times (-4) = -64 ✓ and 3^3 = 27 ✓.

Tip: Cube roots of negative numbers are perfectly fine real numbers — only even roots (square roots, 4th roots) of negatives are not real. Also, find each root separately: \sqrt[3]{a} + \sqrt[3]{b} is not the same as \sqrt[3]{a + b}.

1. Volume of a cube: V = s^3 with s = 2.5 ft.
2. 2.5 \times 2.5 = 6.25.
3. 6.25 \times 2.5 = 15.625.

Answer: 15.625 cubic feet

Check: 2.5 = \tfrac{5}{2}, and \left(\tfrac{5}{2}\right)^3 = \tfrac{125}{8} = 15.625 ✓. Estimate: between 2^3 = 8 and 3^3 = 27 ✓.

Tip: Three similar-looking answers are often offered for a cube: one face (s^2), the surface area (6s^2) and the volume (s^3). Volume is in cubic units and uses three factors of the edge.

Test each choice.

1. 16 = 4^2 is a perfect square. Cubes near it: 2^3 = 8, 3^3 = 27 — 16 is not a cube. ✗
2. 27 = 3^3 is a perfect cube. Squares near it: 5^2 = 25, 6^2 = 36 — 27 is not a square. ✗
3. 100 = 10^2 is a perfect square. Cubes near it: 4^3 = 64, 5^3 = 125 — not a cube. ✗
4. 729 = 27^2 (since 27 \times 27 = 729) and 729 = 9^3 (since 9 \times 9 \times 9 = 729). ✓

Answer: 729

Check: 27^2 = 729 ✓ and 9^3 = 81 \times 9 = 729 ✓. In fact 729 = 3^6, and any number to the 6th power is both a square and a cube.

Tip: Numbers that are both squares and cubes are sixth powers: 1, 64, 729, 4{,}096, \ldots (that is 1^6, 2^6, 3^6, 4^6).

1. List perfect cubes: 1^3 = 1, 2^3 = 8, 3^3 = 27, 4^3 = 64, 5^3 = 125.
2. 64 < 100 < 125.
3. So \sqrt[3]{64} < \sqrt[3]{100} < \sqrt[3]{125}, which means 4 < \sqrt[3]{100} < 5.

Answer: between 4 and 5

Check: Calculator: \sqrt[3]{100} \approx 4.64 ✓. And 4.64^3 \approx 99.9 ✓.

Tip: Do not mix up roots: \sqrt{100} = 10 but \sqrt[3]{100} \approx 4.6. A cube root is always smaller than a square root for numbers greater than 1.

1. Small box: V = 4^3 = 64 cubic inches.
2. Large box: edge = 3 \times 4 = 12 inches, so V = 12^3 = 1{,}728 cubic inches.
3. Compare: 1{,}728 \div 64 = 27.

Answer: 27 times

Check: Every one of the three dimensions (length, width, height) is 3 times as long, so the volume grows by 3 \times 3 \times 3 = 3^3 = 27 ✓.

Tip: If every edge of a solid is multiplied by k, the volume is multiplied by k^3 (and each face's area by k^2). You do not even need the 4 inches to answer.

Evaluate each choice.

1. \sqrt[3]{-512}: 8^3 = 512, so (-8)^3 = -512 and \sqrt[3]{-512} = -8. ✓
2. -\sqrt[3]{-512} = -(-8) = 8. ✗ (the outside minus flips the sign)
3. \sqrt[3]{-24}: (-2)^3 = -8 and (-3)^3 = -27, so \sqrt[3]{-24} is between -2 and -3 (about -2.88). ✗
4. (-2)^3 \cdot (-1) = (-8)(-1) = 8. ✗

Answer: \sqrt[3]{-512}

Check: (-8)(-8)(-8) = 64 \cdot (-8) = -512 ✓.

Tip: Cube roots of negatives are allowed (unlike square roots): an odd number of negative factors stays negative. Count the minus signs carefully — one inside and one outside cancel out.

1. Edge of the first bin: s^3 = 27, so s = \sqrt[3]{27} = 3 ft.
2. Edge of the second bin: twice as long, 2 \times 3 = 6 ft.
3. Volume of the second bin: 6^3 = 6 \times 6 \times 6 = 216 cubic feet.

Answer: 216 cubic feet

Check: Doubling every edge multiplies the volume by 2^3 = 8: 27 \times 8 = 216 ✓.

Tip: When all lengths are multiplied by k, areas are multiplied by k^2 and volumes by k^3. "Twice as long" means 8 times the volume, not twice.

1. For a cube, s = \sqrt[3]{V} = \sqrt[3]{42.875}.
2. Bracket: 3^3 = 27 and 4^3 = 64, so the edge is between 3 and 4 feet.
3. The volume ends in 5 (like 5^3 = 125), so try 3.5: 3.5 \times 3.5 = 12.25, and 12.25 \times 3.5 = 42.875 ✓.
4. On the TI-30XS: type 3, press 2nd then ^ (for \sqrt[x]{\ }), type 42.875, press enter → 3.5.

Answer: 3.5 feet

Check: 3.5^3 = 42.875 ✓, and 3.5 is between 3 and 4 as the bracket predicted ✓.

Tip: The cube root of a decimal can have fewer decimal places than the number: 42.875 has three places and its cube root 3.5 has one, because 1 + 1 + 1 = 3.

1. Small edge: \sqrt[3]{8} = 2 inches.
2. Large edge: \sqrt[3]{216} = 6 inches (because 6^3 = 216).
3. Compare: 6 \div 2 = 3. The large edge is 3 times as long.

Answer: 3 times as long

Check: The volume ratio is 216 \div 8 = 27, and \sqrt[3]{27} = 3 ✓. Edges scaled by 3 make the volume 3^3 = 27 times as large.

Tip: Volume ratio = (edge ratio)^3, so edge ratio = \sqrt[3]{\text{volume ratio}}. "How many times as long" asks for a ratio (divide), not a difference (subtract).

1. Undo "multiply by \tfrac{1}{2}" by multiplying both sides by 2: x^3 = -216.
2. Take the cube root: x = \sqrt[3]{-216}.
3. 6^3 = 216, so (-6)^3 = -216. Therefore x = -6.

Answer: -6

Check: \tfrac{1}{2}(-6)^3 = \tfrac{1}{2}(-216) = -108 ✓.

Tip: Cube roots of negative numbers are negative, and a cubic equation like this has only one real solution. Don't drop the negative sign!

1. Cube: \left(-\tfrac{2}{3}\right)^3 = \dfrac{(-2)^3}{3^3} = \dfrac{-8}{27} = -\dfrac{8}{27} (three negative factors → negative).
2. Cube root: \sqrt[3]{\tfrac{1}{27}} = \dfrac{\sqrt[3]{1}}{\sqrt[3]{27}} = \dfrac{1}{3}.
3. Add: -\dfrac{8}{27} + \dfrac{1}{3} = -\dfrac{8}{27} + \dfrac{9}{27} = \dfrac{1}{27}.

Answer: \dfrac{1}{27}

Check: Decimals: -\tfrac{8}{27} \approx -0.296 and \tfrac{1}{3} \approx 0.333; sum \approx 0.037, and \tfrac{1}{27} \approx 0.037 ✓.

Tip: When you raise a fraction to a power, apply the power to both the numerator and the denominator. The same goes for roots.

Test each statement.

1. \sqrt[3]{64} = 4 (because 4^3 = 64) and \sqrt{16} = 4 (because 4^2 = 16). Both sides are 4. True.
2. \sqrt[3]{27} + \sqrt[3]{1} = 3 + 1 = 4, but \sqrt[3]{28} \approx 3.04. False — roots do not split over addition.
3. (-3)^3 = (-3)(-3)(-3) = -27, not 27. False.
4. \left(\sqrt[3]{8}\right)^2 = 2^2 = 4, not 8. False — a cube root is undone by cubing, not by squaring.

Answer: \sqrt[3]{64} = \sqrt{16}

Check: 4^3 = 64 ✓ and 4^2 = 16 ✓.

Tip: Squaring undoes a square root, and cubing undoes a cube root. Mixing them up (squaring a cube root) does not get you back where you started.

1. Edge: s = \sqrt[3]{3.375}. Bracket: 1^3 = 1 and 2^3 = 8, so s is between 1 and 2. Try 1.5: 1.5 \times 1.5 = 2.25, 2.25 \times 1.5 = 3.375 ✓. So s = 1.5 feet.
2. One face: a square, 1.5 \times 1.5 = 2.25 square feet.
3. Six faces: 6 \times 2.25 = 13.5 square feet.

Answer: 13.5 square feet

Check: Work backwards: 13.5 \div 6 = 2.25 per face; \sqrt{2.25} = 1.5 ft edge; 1.5^3 = 3.375 cubic feet ✓.

Tip: Volume → edge needs a cube root; edge → face area needs a square. Keep track of units: cubic feet for volume, square feet for paint.

A counterexample must be a perfect cube that is not a perfect square.

1. 1 = 1^3 = 1^2: a cube and a square. It agrees with Rosa. ✗
2. 8 = 2^3 is a perfect cube. Squares near it: 2^2 = 4 and 3^2 = 9, so 8 is not a perfect square. This disproves the claim. ✓
3. 36 = 6^2 is a perfect square but not a cube (3^3 = 27, 4^3 = 64). Since it is not a cube, it says nothing about Rosa's claim. ✗
4. 64 = 4^3 = 8^2: both a cube and a square. It agrees with Rosa. ✗

Answer: 8

Check: 2 \times 2 \times 2 = 8 ✓, and no whole number times itself equals 8 (2 \times 2 = 4, 3 \times 3 = 9) ✓.

Tip: A counterexample to "every A is B" must be an A that is not B. A B that is not an A (like 36) does not disprove anything.