Question

(1)What is the smallest number by which 3087 must be multiplied so that product is a perfect cube? What will be the cube root of the number obtained?
(2)What is the length of a cube whose volume is 2187m3?
(3)Is (6,8,10) a Pythagorean triplet? Explain your answer?
(4)Evaluate √2 correct up to two decimal places?
(5)Find the cube of 2.5?

Answer 1

Answer:

1. To find the smallest number by which 3087 must be multiplied so that the product is a perfect cube, we need to factorize 3087 into its prime factors.

3087 = 3 × 3 × 3 × 7 × 13

To make this number a perfect cube, we need to multiply it by 3 × 7 × 13, which is the smallest number that will make all the exponents of the prime factors a multiple of 3.

So, 3087 × 3 × 7 × 13 = 3³ × 7³ × 13³ = (3 × 7 × 13)³

Therefore, the smallest number by which 3087 must be multiplied so that the product is a perfect cube is 3 × 7 × 13 = 273, and the cube root of the number obtained is 3 × 7 × 13 = 273.

2. The volume of a cube is given by the formula V = s³, where s is the length of a side of the cube.

We are given that the volume of the cube is 2187 m³.

So, 2187 = s³

Taking the cube root of both sides, we get:

s = ³√2187

Simplifying, we get:

s = 9

Therefore, the length of the cube is 9 m.

3. Yes, (6, 8, 10) is a Pythagorean triplet.

A Pythagorean triplet is a set of three positive integers a, b, and c that satisfy the equation a² + b² = c².

In this case, we have:

6² + 8² = 36 + 64 = 100

10² = 100

So, 6² + 8² = 10², which means that (6, 8, 10) is a Pythagorean triplet.

4. To evaluate √2 correct up to two decimal places, we can use a calculator or approximation method.

Using a calculator, we get:

√2 = 1.41421356...

Rounding to two decimal places, we get:

√2 ≈ 1.41

Therefore, √2 correct up to two decimal places is 1.41.

5. The cube of 2.5 is:

(2.5)³ = 2.5 ×