To express each given number as a product of prime factors, we will break down each number into its prime components step-by-step.
### (a) Factorize 48
We start by dividing 48 by the smallest prime number, which is 2:
1. [tex]\( 48 \div 2 = 24 \)[/tex]
2. [tex]\( 24 \div 2 = 12 \)[/tex]
3. [tex]\( 12 \div 2 = 6 \)[/tex]
4. [tex]\( 6 \div 2 = 3 \)[/tex]
5. 3 is a prime number.
So, [tex]\( 48 = 2^4 \times 3 \)[/tex].
### (b) Factorize 160
Similarly, we divide 160 by 2, the smallest prime:
1. [tex]\( 160 \div 2 = 80 \)[/tex]
2. [tex]\( 80 \div 2 = 40 \)[/tex]
3. [tex]\( 40 \div 2 = 20 \)[/tex]
4. [tex]\( 20 \div 2 = 10 \)[/tex]
5. [tex]\( 10 \div 2 = 5 \)[/tex]
6. 5 is a prime number.
Hence, [tex]\( 160 = 2^5 \times 5 \)[/tex].
### (c) Factorize 182
For 182, we start with the smallest prime number, 2:
1. [tex]\( 182 \div 2 = 91 \)[/tex]
2. Now 91 is checked for smaller primes: [tex]\( 91 \div 7 = 13 \)[/tex]
3. 13 is a prime number.
Thus, [tex]\( 182 = 2 \times 7 \times 13 \)[/tex].
### (d) Factorize 512
Lastly, we factorize 512 by repeatedly dividing it by 2:
1. [tex]\( 512 \div 2 = 256 \)[/tex]
2. [tex]\( 256 \div 2 = 128 \)[/tex]
3. [tex]\( 128 \div 2 = 64 \)[/tex]
4. [tex]\( 64 \div 2 = 32 \)[/tex]
5. [tex]\( 32 \div 2 = 16 \)[/tex]
6. [tex]\( 16 \div 2 = 8 \)[/tex]
7. [tex]\( 8 \div 2 = 4 \)[/tex]
8. [tex]\( 4 \div 2 = 2 \)[/tex]
9. [tex]\( 2 \div 2 = 1 \)[/tex]
Therefore, [tex]\( 512 = 2^9 \)[/tex].
### Summary
Here are the factorizations of the numbers:
- [tex]\( 48 = 2^4 \times 3 \)[/tex]
- [tex]\( 160 = 2^5 \times 5 \)[/tex]
- [tex]\( 182 = 2 \times 7 \times 13 \)[/tex]
- [tex]\( 512 = 2^9 \)[/tex]
