Cube Root Worksheet-2 from Learners' Planet, featuring math problems on evaluating cube roots, filling gaps, and showing calculations.

Worksheet-2: Cube Root

#### Long Question: Evaluate the following

We will solve each problem step by step.

---

#### (1) \( \left\{ (5^2 + 12^2)^{1/2} \right\}^3 \)

1. Calculate \( 5^2 \):
\[
5^2 = 25
\]

2. Calculate \( 12^2 \):
\[
12^2 = 144
\]

3. Add \( 5^2 \) and \( 12^2 \):
\[
5^2 + 12^2 = 25 + 144 = 169
\]

4. Take the square root of 169:
\[
\sqrt{169} = 13
\]

5. Cube the result:
\[
13^3 = 13 \times 13 \times 13 = 2197
\]

Answer:
\[
\boxed{2197}
\]

---

#### (2) \( \left\{ (6^2 + 8^2)^{1/2} \right\}^3 \)

1. Calculate \( 6^2 \):
\[
6^2 = 36
\]

2. Calculate \( 8^2 \):
\[
8^2 = 64
\]

3. Add \( 6^2 \) and \( 8^2 \):
\[
6^2 + 8^2 = 36 + 64 = 100
\]

4. Take the square root of 100:
\[
\sqrt{100} = 10
\]

5. Cube the result:
\[
10^3 = 10 \times 10 \times 10 = 1000
\]

Answer:
\[
\boxed{1000}
\]

---

#### (3) \( \sqrt[3]{4^3 \times 6^3} \)

1. Simplify the expression inside the cube root:
\[
4^3 \times 6^3 = (4 \times 6)^3 = 24^3
\]

2. Take the cube root:
\[
\sqrt[3]{24^3} = 24
\]

Answer:
\[
\boxed{24}
\]

---

#### (4) \( \sqrt[3]{8 \times 17 \times 17 \times 17} \)

1. Rewrite the expression:
\[
\sqrt[3]{8 \times 17^3}
\]

2. Separate the cube root:
\[
\sqrt[3]{8} \times \sqrt[3]{17^3} = 2 \times 17 = 34
\]

Answer:
\[
\boxed{34}
\]

---

#### (5) \( \sqrt[3]{700 \times 2 \times 49 \times 5} \)

1. Simplify the expression inside the cube root:
\[
700 \times 2 \times 49 \times 5 = (7 \times 100) \times 2 \times 49 \times 5 = 7 \times 2 \times 49 \times (100 \times 5) = 7 \times 2 \times 49 \times 500
\]

2. Factorize further:
\[
7 \times 2 \times 49 \times 500 = 7 \times 2 \times 7^2 \times (5^3 \times 4) = 7^3 \times 2^2 \times 5^3
\]

3. Take the cube root:
\[
\sqrt[3]{7^3 \times 2^2 \times 5^3} = \sqrt[3]{7^3} \times \sqrt[3]{5^3} \times \sqrt[3]{2^2} = 7 \times 5 \times \sqrt[3]{4} = 35 \sqrt[3]{4}
\]

Answer:
\[
\boxed{35 \sqrt[3]{4}}
\]

---

#### (6) \( 125 \sqrt[3]{a^6} - \sqrt[3]{125 a^6} \)

1. Simplify \( \sqrt[3]{a^6} \):
\[
\sqrt[3]{a^6} = a^2
\]

2. Substitute back:
\[
125 \sqrt[3]{a^6} - \sqrt[3]{125 a^6} = 125 a^2 - \sqrt[3]{125 a^6}
\]

3. Simplify \( \sqrt[3]{125 a^6} \):
\[
\sqrt[3]{125 a^6} = \sqrt[3]{125} \times \sqrt[3]{a^6} = 5 \times a^2 = 5a^2
\]

4. Substitute back:
\[
125 a^2 - 5a^2 = 120a^2
\]

Answer:
\[
\boxed{120a^2}
\]

---

#### (7) \( \sqrt[3]{27} + \sqrt[3]{0.008} + \sqrt[3]{0.064} \)

1. Simplify each cube root:
\[
\sqrt[3]{27} = 3, \quad \sqrt[3]{0.008} = 0.2, \quad \sqrt[3]{0.064} = 0.4
\]

2. Add the results:
\[
3 + 0.2 + 0.4 = 3.6
\]

Answer:
\[
\boxed{3.6}
\]

---

#### (8) \( \sqrt[3]{1000} + \sqrt[3]{0.008} - \sqrt[3]{0.125} \)

1. Simplify each cube root:
\[
\sqrt[3]{1000} = 10, \quad \sqrt[3]{0.008} = 0.2, \quad \sqrt[3]{0.125} = 0.5
\]

2. Perform the operations:
\[
10 + 0.2 - 0.5 = 9.7
\]

Answer:
\[
\boxed{9.7}
\]

---

#### (9) \( \sqrt[3]{\frac{729}{216}} \times \frac{6}{9} \)

1. Simplify the fraction inside the cube root:
\[
\frac{729}{216} = \frac{27^3}{6^3} = \left( \frac{27}{6} \right)^3 = \left( \frac{9}{2} \right)^3
\]

2. Take the cube root:
\[
\sqrt[3]{\frac{729}{216}} = \sqrt[3]{\left( \frac{9}{2} \right)^3} = \frac{9}{2}
\]

3. Multiply by \( \frac{6}{9} \):
\[
\frac{9}{2} \times \frac{6}{9} = \frac{9 \times 6}{2 \times 9} = \frac{6}{2} = 3
\]

Answer:
\[
\boxed{3}
\]

---

#### (10) \( \sqrt[3]{0.1 \times 0.1 \times 0.1 \times 13 \times 13 \times 13} \)

1. Simplify the expression inside the cube root:
\[
0.1 \times 0.1 \times 0.1 = (0.1)^3 = 0.001
\]
\[
13 \times 13 \times 13 = 13^3
\]
\[
0.1 \times 0.1 \times 0.1 \times 13 \times 13 \times 13 = 0.001 \times 13^3
\]

2. Take the cube root:
\[
\sqrt[3]{0.001 \times 13^3} = \sqrt[3]{0.001} \times \sqrt[3]{13^3} = 0.1 \times 13 = 1.3
\]

Answer:
\[
\boxed{1.3}
\]

---

#### (11) \( \sqrt[3]{36} \times \sqrt[3]{384} \)

1. Combine the cube roots:
\[
\sqrt[3]{36} \times \sqrt[3]{384} = \sqrt[3]{36 \times 384}
\]

2. Simplify \( 36 \times 384 \):
\[
36 \times 384 = 13824
\]

3. Factorize 13824:
\[
13824 = 2^9 \times 3^3
\]

4. Take the cube root:
\[
\sqrt[3]{13824} = \sqrt[3]{2^9 \times 3^3} = 2^3 \times 3 = 8 \times 3 = 24
\]

Answer:
\[
\boxed{24}
\]

---

#### (12) \( \sqrt[3]{96} \times \sqrt[3]{144} \)

1. Combine the cube roots:
\[
\sqrt[3]{96} \times \sqrt[3]{144} = \sqrt[3]{96 \times 144}
\]

2. Simplify \( 96 \times 144 \):
\[
96 \times 144 = 13824
\]

3. Factorize 13824:
\[
13824 = 2^9 \times 3^3
\]

4. Take the cube root:
\[
\sqrt[3]{13824} = \sqrt[3]{2^9 \times 3^3} = 2^3 \times 3 = 8 \times 3 = 24
\]

Answer:
\[
\boxed{24}
\]

---

#### (13) \( \sqrt[3]{121} \times \sqrt[3]{297} \)

1. Combine the cube roots:
\[
\sqrt[3]{121} \times \sqrt[3]{297} = \sqrt[3]{121 \times 297}
\]

2. Simplify \( 121 \times 297 \):
\[
121 \times 297 = 35937
\]

3. Factorize 35937:
\[
35937 = 3^9 \times 11
\]

4. Take the cube root:
\[
\sqrt[3]{35937} = \sqrt[3]{3^9 \times 11} = 3^3 \times \sqrt[3]{11} = 27 \sqrt[3]{11}
\]

Answer:
\[
\boxed{27 \sqrt[3]{11}}
\]

---

#### (14) \( \sqrt[3]{100} \times \sqrt[3]{270} \)

1. Combine the cube roots:
\[
\sqrt[3]{100} \times \sqrt[3]{270} = \sqrt[3]{100 \times 270}
\]

2. Simplify \( 100 \times 270 \):
\[
100 \times 270 = 27000
\]

3. Factorize 27000:
\[
27000 = 3^3 \times 10^3
\]

4. Take the cube root:
\[
\sqrt[3]{27000} = \sqrt[3]{3^3 \times 10^3} = 3 \times 10 = 30
\]

Answer:
\[
\boxed{30}
\]

---

Fill the gap:

#### (15) \( \sqrt[3]{125 \times 27} = 3 \times \ldots \)

1. Simplify the left-hand side:
\[
\sqrt[3]{125 \times 27} = \sqrt[3]{125} \times \sqrt[3]{27} = 5 \times 3 = 15
\]

2. The equation becomes:
\[
15 = 3 \times \ldots
\]

3. Solve for the missing term:
\[
\ldots = \frac{15}{3} = 5
\]

Answer:
\[
\boxed{5}
\]

---

#### (16) \( \sqrt[3]{8 \times \ldots} = 8 \)

1. Let the missing term be \( x \). The equation becomes:
\[
\sqrt[3]{8 \times x} = 8
\]

2. Cube both sides:
\[
8 \times x = 8^3 = 512
\]

3. Solve for \( x \):
\[
x = \frac{512}{8} = 64
\]

Answer:
\[
\boxed{64}
\]

---

#### (17) \( \sqrt[3]{1728} = 4 \times \ldots \)

1. Simplify the left-hand side:
\[
\sqrt[3]{1728} = 12
\]

2. The equation becomes:
\[
12 = 4 \times \ldots
\]

3. Solve for the missing term:
\[
\ldots = \frac{12}{4} = 3
\]

Answer:
\[
\boxed{3}
\]

---

#### (18) \( \sqrt[3]{480} = \sqrt[3]{3} \times 2 \times \sqrt[3]{\ldots} \)

1. Factorize 480:
\[
480 = 2^5 \times 3 \times 5
\]

2. Rewrite the cube root:
\[
\sqrt[3]{480} = \sqrt[3]{2^5 \times 3 \times 5} = \sqrt[3]{2^3 \times 2^2 \times 3 \times 5} = 2 \sqrt[3]{4 \times 3 \times 5} = 2 \sqrt[3]{60}
\]

3. The equation becomes:
\[
\sqrt[3]{480} = \sqrt[3]{3} \times 2 \times \sqrt[3]{\ldots}
\]

4. Compare:
\[
\sqrt[3]{60} = \sqrt[3]{3} \times \sqrt[3]{\ldots}
\]

5. Solve for the missing term:
\[
\ldots = \frac{60}{3} = 20
\]

Answer:
\[
\boxed{20}
\]

---

#### (19) \( \sqrt[3]{\ldots} = \sqrt[3]{7} \times \sqrt[3]{8} \)

1. Simplify the right-hand side:
\[
\sqrt[3]{7} \times \sqrt[3]{8} = \sqrt[3]{7 \times 8} = \sqrt[3]{56}
\]

2. The equation becomes:
\[
\sqrt[3]{\ldots} = \sqrt[3]{56}
\]

3. Solve for the missing term:
\[
\ldots = 56
\]

Answer:
\[
\boxed{56}
\]

---

#### (20) \( \sqrt[3]{\ldots} = \sqrt[3]{4} \times \sqrt[3]{5} \times \sqrt[3]{6} \)

1. Simplify the right-hand side:
\[
\sqrt[3]{4} \times \sqrt[3]{5} \times \sqrt[3]{6} = \sqrt[3]{4 \times 5 \times 6} = \sqrt[3]{120}
\]

2. The equation becomes:
\[
\sqrt[3]{\ldots} = \sqrt[3]{120}
\]

3. Solve for the missing term:
\[
\ldots = 120
\]

Answer:
\[
\boxed{120}
\]

---

Show that:

#### (24) \( \sqrt[3]{27} \times \sqrt[3]{64} = \sqrt[3]{27 \times 64} \)

1. Simplify the left-hand side:
\[
\sqrt[3]{27} = 3, \quad \sqrt[3]{64} = 4
\]
\[
\sqrt[3]{27} \times \sqrt[3]{64} = 3 \times 4 = 12
\]

2. Simplify the right-hand side:
\[
27 \times 64 = 1728
\]
\[
\sqrt[3]{27 \times 64} = \sqrt[3]{1728} = 12
\]

3. Both sides are equal:
\[
12 = 12
\]

Answer:
\[
\boxed{\text{Proven}}
\]

---

#### (25) \( \sqrt[3]{64 \times 729} = \sqrt[3]{64} \times \sqrt[3]{729} \)

1. Simplify the left-hand side:
\[
64 \times 729 = 46656
\]
\[
\sqrt[3]{64 \times 729} = \sqrt[3]{46656} = 36
\]

2. Simplify the right-hand side:
\[
\sqrt[3]{64} = 4, \quad \sqrt[3]{729} = 9
\]
\[
\sqrt[3]{64} \times \sqrt[3]{729} = 4 \times 9 = 36
\]

3. Both sides are equal:
\[
36 = 36
\]

Answer:
\[
\boxed{\text{Proven}}
\]

---

#### (26) \( \sqrt[3]{-125 \times 216} = \sqrt[3]{-125} \times \sqrt[3]{216} \)

1. Simplify the left-hand side:
\[
-125 \times 216 = -27000
\]
\[
\sqrt[3]{-125 \times 216} = \sqrt[3]{-27000} = -30
\]

2. Simplify the right-hand side:
\[
\sqrt[3]{-125} = -5, \quad \sqrt[3]{216} = 6
\]
\[
\sqrt[3]{-125} \times \sqrt[3]{216} = -5 \times 6 = -30
\]

3. Both sides are equal:
\[
-30 = -30
\]

Answer:
\[
\boxed{\text{Proven}}
\]

---

Final Answers:

\[
\boxed{
\begin{array}{ll}
(1) & 2197 \\
(2) & 1000 \\
(3) & 24 \\
(4) & 34 \\
(5) & 35 \sqrt[3]{4} \\
(6) & 120a^2 \\
(7) & 3.6 \\
(8) & 9.7 \\
(9) & 3 \\
(10) & 1.3 \\
(11) & 24 \\
(12) & 24 \\
(13) & 27 \sqrt[3]{11} \\
(14) & 30 \\
(15) & 5 \\
(16) & 64 \\
(17) & 3 \\
(18) & 20 \\
(19) & 56 \\
(20) & 120 \\
(24) & \text{Proven} \\
(25) & \text{Proven} \\
(26) & \text{Proven} \\
\end{array}
}
\]