1.   Is 9000 a perfect cube?
2.   By which smallest number should 42592 be divided so that the quotient is a perfect cube?
3.   Show that 46656 is a perfect cube.
4.   By which smallest number should 704 be divided to obtain a perfect cube?
5.   Find the cue root of 9197.
6.   Show that 384 is not a perfect cube.
7.   By which smallest number should 648 be multiplied so that the product is a perfect cube?
8.   Find the number whose cube is 27000.

Let's go step by step to answer each of the questions.

### 1. **Is 9000 a perfect cube?**

A perfect cube is a number that can be expressed as \( n^3 \), where \( n \) is an integer.

Factorizing 9000:
\[
9000 = 2^3 \times 3^2 \times 5^3
\]
For a number to be a perfect cube, all prime factors must appear in powers that are multiples of 3. Here, the factor \( 3^2 \) doesn't meet this requirement because the exponent is not divisible by 3.

Thus, **9000 is not a perfect cube**.

---

### 2. **By which smallest number should 42592 be divided so that the quotient is a perfect cube?**

First, let's factorize 42592:
\[
42592 = 2^8 \times 11^3
\]
For a number to be a perfect cube, each prime factor should have an exponent divisible by 3. Here, \( 2^8 \) has an exponent of 8, which is not divisible by 3. To make it divisible by 3, we need to divide by \( 2^2 \), the smallest factor that reduces the exponent of 2 to 6 (which is divisible by 3).

Thus, **42592 should be divided by 4** (since \( 2^2 = 4 \)) to make the quotient a perfect cube.

---

### 3. **Show that 46656 is a perfect cube.**

First, factorize 46656:
\[
46656 = 2^6 \times 3^6
\]
Since all exponents (6 for both 2 and 3) are divisible by 3, \( 46656 \) is indeed a perfect cube.

The cube root of 46656 is:
\[
\sqrt[3]{46656} = \sqrt[3]{2^6 \times 3^6} = 2^2 \times 3^2 = 4 \times 9 = 36
\]

Thus, **46656 is a perfect cube**, and its cube root is 36.

---

### 4. **By which smallest number should 704 be divided to obtain a perfect cube?**

Factorizing 704:
\[
704 = 2^6 \times 11
\]
For 704 to become a perfect cube, all prime factors need to have exponents divisible by 3. The prime factor \( 11 \) has an exponent of 1, which is not divisible by 3. To make 704 a perfect cube, we need to divide by 11 to remove this factor.

Thus, **704 should be divided by 11** to obtain a perfect cube.

---

### 5. **Find the cube root of 9197.**

Let's approximate the cube root of 9197.

The cube root of 9197 is approximately:
\[
\sqrt[3]{9197} \approx 20.95
\]

Thus, the cube root of 9197 is approximately **20.95**.

---

### 6. **Show that 384 is not a perfect cube.**

Factorizing 384:
\[
384 = 2^7 \times 3
\]
For a number to be a perfect cube, all prime factors must have exponents divisible by 3. The exponents of both 2 and 3 are not divisible by 3, which shows that **384 is not a perfect cube**.

---

### 7. **By which smallest number should 648 be multiplied so that the product is a perfect cube?**

Factorizing 648:
\[
648 = 2^3 \times 3^4
\]
The factor \( 3^4 \) has an exponent of 4, which is not divisible by 3. To make the exponent divisible by 3, we need to multiply by \( 3^2 \) (which is 9) to increase the exponent of 3 to 6.

Thus, **648 should be multiplied by 9** to make the product a perfect cube.

---

### 8. **Find the number whose cube is 27000.**

We need to find \( n \) such that:
\[
n^3 = 27000
\]
The cube root of 27000 is:
\[
\sqrt[3]{27000} = \sqrt[3]{3^3 \times 10^3} = 30
\]

Thus, the number whose cube is 27000 is **30**.

---

Would you like more details or have any questions on these problems?

### Here are 5 related questions for further practice:
1. What is the smallest number by which 360 should be divided to obtain a perfect cube?
2. Find the smallest number that should be multiplied by 500 to make it a perfect cube.
3. Is 512 a perfect cube? If yes, what is its cube root?
4. Show that 128 is not a perfect cube.
5. What is the cube root of 3375?

### Tip:
When checking if a number is a perfect cube, always factorize the number and check if all the exponents in its prime factorization are divisible by 3.

1. Is 9000 a perfect cube?
2. By which smallest number should 42592 be divided so that the quotient is a perfect cube?
3. Show that 46656 is a perfect cube.
4. By which smallest number should 704 be divided to obtain a perfect cube?
5. Find the cube root of 9197.
6. Show that 384 is not a perfect cube.
7. By which smallest number should 648 be multiplied so that the product is a perfect cube?
8. Find the number whose cube is 27000.

Solve math problems related to perfect cubes, including checking if numbers like 9000 and 46656 are perfect cubes, finding cube roots, and determining divisors or multipliers to make numbers perfect cubes. Learn with step-by-step explanations suitable for Grades 6-8.

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