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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.