Cube Roots

Worked example

1. 1

   $4\times 4=16$ — Start with 4. Multiply the first two fours, because two at a time is easier.

2. 2

   $16\times 4=64$ — Multiply by the third 4. So 4 cubed is 64.

3. 3

   $\sqrt[3]{64}=4$ — The cube root goes backwards: 4 is the number that makes 64 when you use it three times.

Worked example

1. 1

   $(-2)\times (-2)=4$ — Minus times minus is plus, so the first two give +4.

2. 2

   $4\times (-2)=-8$ — Plus times minus is minus, so the third −2 makes the answer negative.

3. 3

   $\sqrt[3]{-8}=-2$ — Three negatives multiply to a negative, so the cube root of a negative number is negative.

Worked example

1. 1

   $10^3=1000,20^3=8000$ — Estimate first. 4913 is between 1000 and 8000, so its cube root is between 10 and 20.

2. 2

   $\sqrt[3]{4913}=17$ — Press the cube root key, then type 4913. 17 is between 10 and 20, so it fits the estimate.

3. 3

   $17\times 17=289$ — Now check by cubing back. Multiply the first two 17s.

4. 4

   $289\times 17=4913$ — Multiply by the third 17. It gives 4913, the number you started with, so 17 is right.

Worked example

1. 1

   $4^3=64,5^3=125$ — Estimate first. 91.125 is between 64 and 125, so its cube root is between 4 and 5.

2. 2

   $\sqrt[3]{91.125}=4.5$ — Cube root key. 4.5 is between 4 and 5, so the decimal point is in the right place.

3. 3

   $4.5\times 4.5=20.25$ — Check by cubing back. Multiply the first two.

4. 4

   $20.25\times 4.5=91.125$ — Multiply by the third 4.5. It matches 91.125, so 4.5 is right.

Worked example

1. 1

   $5000^{\frac{1}{3}}=\sqrt[3]{5000}$ — Rewrite the power as a cube root, so you know what you are finding.

2. 2

   $17^3=4913,18^3=5832$ — Estimate first. 5000 is between these, so the answer is between 17 and 18.

3. 3

   $\sqrt[3]{5000}=17.09975947$ — Use the cube root key and write down the full display. This line can earn a mark by itself, even if the rounding goes wrong.

4. 4

   $17.09975947$ — For 2 decimal places, look at the 3rd decimal. It is 9, which is 5 or more, so round up.

5. 5

   $\approx 17.10$ — 17.09 rounds up to 17.10. Keep the 0: 2 decimal places means two digits after the point.

Worked example

1. 1

   $\sqrt[3]{?}=12$ — 12 is the side of the cube. The missing number is the whole cube, so it is 12 cubed.

2. 2

   $12\times 12=144$ — Cube two at a time. Multiply the first two 12s.

3. 3

   $144\times 12=1728$ — Multiply by the third 12. So the missing number is 1728.

4. 4

   $\sqrt[3]{1728}=12$ — Check with the cube root key. It gives back 12, so 1728 is right.

Worked example

1. 1

   $\sqrt[3]{\frac{8}{27}}=\frac{\sqrt[3]{8}}{\sqrt[3]{27}}$ — Split into the top and the bottom. Both are cube numbers.

2. 2

   $\sqrt[3]{8}=2$ — 2 × 2 × 2 = 8, so the cube root of the top is 2.

3. 3

   $\sqrt[3]{27}=3$ — 3 × 3 × 3 = 27, so the cube root of the bottom is 3.

4. 4

   $\sqrt[3]{\frac{8}{27}}=\frac{2}{3}$ — Put them back together. Check: $\frac{2}{3}\times \frac{2}{3}\times \frac{2}{3}=\frac{8}{27}$.

Worked example

1. 1

   $0.064=\frac{64}{1000}$ — 0.064 has 3 decimal places, so it is 64 thousandths. 1000 is a cube: 10 × 10 × 10.

2. 2

   $\sqrt[3]{\frac{64}{1000}}=\frac{4}{10}$ — Cube root of the top is 4, because 4 × 4 × 4 = 64. Cube root of the bottom is 10.

3. 3

   $\frac{4}{10}=0.4$ — Change back to a decimal. Check: 0.4 × 0.4 = 0.16, and 0.16 × 0.4 = 0.064.

Worked example

1. 1

   $l^3=512$ — Volume of a cube = side × side × side. Put in the volume, 512.

2. 2

   $l=\sqrt[3]{512}$ — Take the cube root of both sides, to undo the cube and leave the side alone.

3. 3

   $l=8$ — 8 × 8 × 8 = 512, so the side is 8.

4. 4

   $8\times 8=64,64\times 8=512$ — Check by cubing back, two at a time. It matches the volume.

5. 5

   $l=8\text{ cm}$ — The volume is in cm³, so the side is in cm. Cubic units become plain length units.