Let's solve the problem step-by-step:
### Step 1: Determine the Prime Factorization of 1536
To start, we need to find the prime factors of 1536:
- 1536 can be divided by 2 repeatedly:
- 1536 ÷ 2 = 768
- 768 ÷ 2 = 384
- 384 ÷ 2 = 192
- 192 ÷ 2 = 96
- 96 ÷ 2 = 48
- 48 ÷ 2 = 24
- 24 ÷ 2 = 12
- 12 ÷ 2 = 6
- 6 ÷ 2 = 3
- Finally, 3 is a prime factor that can no longer be divided by 2.
Therefore, the prime factorization of 1536 is:
[tex]\[ 2^9 \times 3 \][/tex]
### Step 2: Ensure Each Prime Factor's Exponent is a Multiple of 3
We need to make the number a perfect cube. For this, the exponent of each prime factor in its prime factorization needs to be a multiple of 3.
- The prime factor [tex]\(2\)[/tex] has an exponent of 9, which is already a multiple of 3.
- The prime factor [tex]\(3\)[/tex] has an exponent of 1, which is not a multiple of 3. We need to increase the exponent of 3 to the nearest multiple of 3, which is 3. Hence, we need two more 3’s.
### Step 3: Find the Required Multiplier
To make [tex]\(3^1\)[/tex] into [tex]\(3^3\)[/tex], we need to multiply 1536 by:
[tex]\[ 3^2 = 9 \][/tex]
### Step 4: Calculate the Perfect Cube
Multiplying 1536 by 9, we get:
[tex]\[ 1536 \times 9 = 13824 \][/tex]
### Step 5: Calculate the Cube Root of the Perfect Cube
Finally, we find the cube root of 13824:
[tex]\[ \sqrt[3]{13824} = 24 \][/tex]
So, in conclusion:
1. The smallest number by which 1536 should be multiplied to make it a perfect cube is 9.
2. The cube root of the perfect cube (13824) is 24.
