Square & Cube Roots - Go Teach Maths: Handcrafted Resources for - Free Printable

Let's solve each question from the worksheet "Cubes and Cube Roots" step by step with clear explanations.

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1. Find the cube of $\left(-\frac{2}{3}\right)$.

To find the cube of a number, we multiply it by itself three times.

$$
\left(-\frac{2}{3}\right)^3 = \left(-\frac{2}{3}\right) \times \left(-\frac{2}{3}\right) \times \left(-\frac{2}{3}\right)
$$

First:
$$
\left(-\frac{2}{3}\right) \times \left(-\frac{2}{3}\right) = \frac{4}{9}
$$

Then:
$$
\frac{4}{9} \times \left(-\frac{2}{3}\right) = -\frac{8}{27}
$$

✔ Answer: $-\frac{8}{27}$

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2. How much is $\left(\frac{3}{5}\right)^3 - \left(\frac{1}{5}\right)^3$?

We compute each cube:

$$
\left(\frac{3}{5}\right)^3 = \frac{27}{125}, \quad \left(\frac{1}{5}\right)^3 = \frac{1}{125}
$$

Now subtract:

$$
\frac{27}{125} - \frac{1}{125} = \frac{26}{125}
$$

✔ Answer: $\frac{26}{125}$

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3. Simplify and give the answer: $(0.5)^3 + (0.2)^3$

Convert decimals to fractions:

- $0.5 = \frac{1}{2}$ → $(0.5)^3 = \left(\frac{1}{2}\right)^3 = \frac{1}{8}$
- $0.2 = \frac{1}{5}$ → $(0.2)^3 = \left(\frac{1}{5}\right)^3 = \frac{1}{125}$

Now add:

$$
\frac{1}{8} + \frac{1}{125}
$$

Find LCM of 8 and 125: $LCM(8, 125) = 1000$

$$
\frac{1}{8} = \frac{125}{1000}, \quad \frac{1}{125} = \frac{8}{1000}
$$

$$
\frac{125}{1000} + \frac{8}{1000} = \frac{133}{1000} = 0.133
$$

✔ Answer: $0.133$ or $\frac{133}{1000}$

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4. Find the volume of a cube of side 6 cm.

Volume of a cube = $side^3$

$$
6^3 = 6 \times 6 \times 6 = 216 \text{ cm}^3
$$

✔ Answer: $216 \text{ cm}^3$

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5. How many unit cubes can be formed by melting a copper cube of side 5 cm?

A unit cube has side 1 cm.

Volume of original cube = $5^3 = 125 \text{ cm}^3$

Each unit cube has volume $1 \text{ cm}^3$, so number of unit cubes = $125$

✔ Answer: $125$

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6. Simplify $1 + \left(\frac{3}{5}\right)^3$

We already know:
$$
\left(\frac{3}{5}\right)^3 = \frac{27}{125}
$$

So:
$$
1 + \frac{27}{125} = \frac{125}{125} + \frac{27}{125} = \frac{152}{125}
$$

✔ Answer: $\frac{152}{125}$

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7. If a number is written as $3 \times 3 \times 5 \times 3 \times 5 \times 7 \times 7$, find the smallest number by which this is to be multiplied to form a perfect cube.

First, write in prime factorization form:

$$
3 \times 3 \times 3 \times 5 \times 5 \times 7 \times 7 = 3^3 \times 5^2 \times 7^2
$$

For a perfect cube, all exponents must be multiples of 3.

- $3^3$: already good (exponent 3)
- $5^2$: need one more 5 → $5^3$
- $7^2$: need one more 7 → $7^3$

So, multiply by $5 \times 7 = 35$

✔ Answer: $35$

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8. Find $\sqrt[3]{5 \times 7 \times 7 \times 5 \times 7 \times 5}$

Group the factors:

$$
5 \times 5 \times 5 \times 7 \times 7 \times 7 = 5^3 \times 7^3
$$

Now take cube root:

$$
\sqrt[3]{5^3 \times 7^3} = 5 \times 7 = 35
$$

✔ Answer: $35$

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9. $(0.3)^3 = ?$

$$
0.3 = \frac{3}{10}, \quad \left(\frac{3}{10}\right)^3 = \frac{27}{1000} = 0.027
$$

✔ Answer: $0.027$

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10. Find the value of $\sqrt[3]{0.064}$

Note: $0.064 = \frac{64}{1000}$

$$
\sqrt[3]{\frac{64}{1000}} = \frac{\sqrt[3]{64}}{\sqrt[3]{1000}} = \frac{4}{10} = 0.4
$$

Because $4^3 = 64$, $10^3 = 1000$

✔ Answer: $0.4$

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11. $\sqrt[3]{-216} - \sqrt[3]{125} = ?$

$$
\sqrt[3]{-216} = -6 \quad (\text{since } (-6)^3 = -216)
$$
$$
\sqrt[3]{125} = 5 \quad (\text{since } 5^3 = 125)
$$

So:
$$
-6 - 5 = -11
$$

✔ Answer: $-11$

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12. Cubes of positive numbers are always ________.

Positive numbers raised to any odd power (like 3) remain positive.

Example: $2^3 = 8 > 0$, $(-2)^3 = -8 < 0$

So cubes of positive numbers are always positive.

✔ Answer: positive

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13. Cube roots of positive numbers are always ________.

Cube root of a positive number is always positive.

Example: $\sqrt[3]{8} = 2$, $\sqrt[3]{27} = 3$

Even for non-perfect cubes, like $\sqrt[3]{10} \approx 2.15$, still positive.

✔ Answer: positive

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✔ Final Answers Summary:

| Q | Answer |
|---|--------|
| 1 | $-\frac{8}{27}$ |
| 2 | $\frac{26}{125}$ |
| 3 | $0.133$ or $\frac{133}{1000}$ |
| 4 | $216 \text{ cm}^3$ |
| 5 | $125$ |
| 6 | $\frac{152}{125}$ |
| 7 | $35$ |
| 8 | $35$ |
| 9 | $0.027$ |
| 10 | $0.4$ |
| 11 | $-11$ |
| 12 | positive |
| 13 | positive |

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