Alright, let's work through each part of the question step by step.
### 6. Find the bigger number in each of the following pairs.
#### a. [tex]\( 2^5, 5^2 \)[/tex]
First, we calculate each expression:
- [tex]\( 2^5 = 2 \times 2 \times 2 \times 2 \times 2 = 32 \)[/tex]
- [tex]\( 5^2 = 5 \times 5 = 25 \)[/tex]
Comparing the two results, [tex]\( 32 \)[/tex] is greater than [tex]\( 25 \)[/tex].
So, the bigger number in this pair is [tex]\( 2^5 \)[/tex] which is [tex]\( 32 \)[/tex].
#### b. [tex]\( 3^4, 4^3 \)[/tex]
Next, we calculate each expression:
- [tex]\( 3^4 = 3 \times 3 \times 3 \times 3 = 81 \)[/tex]
- [tex]\( 4^3 = 4 \times 4 \times 4 = 64 \)[/tex]
Comparing the two results, [tex]\( 81 \)[/tex] is greater than [tex]\( 64 \)[/tex].
So, the bigger number in this pair is [tex]\( 3^4 \)[/tex] which is [tex]\( 81 \)[/tex].
### 7. Express each of the following numbers as the product of powers of prime factors.
#### a. 324
We need to find the prime factors of 324. Breaking it down, we get:
- [tex]\( 324 \div 2 = 162 \)[/tex]
- [tex]\( 162 \div 2 = 81 \)[/tex]
- [tex]\( 81 \div 3 = 27 \)[/tex]
- [tex]\( 27 \div 3 = 9 \)[/tex]
- [tex]\( 9 \div 3 = 3 \)[/tex]
- [tex]\( 3 \div 3 = 1 \)[/tex]
So, [tex]\( 324 = 2^2 \times 3^4 \)[/tex].
#### b. 1728
For 1728, we break it down as follows:
- [tex]\( 1728 \div 2 = 864 \)[/tex]
- [tex]\( 864 \div 2 = 432 \)[/tex]
- [tex]\( 432 \div 2 = 216 \)[/tex]
- [tex]\( 216 \div 2 = 108 \)[/tex]
- [tex]\( 108 \div 2 = 54 \)[/tex]
- [tex]\( 54 \div 2 = 27 \)[/tex]
- [tex]\( 27 \div 3 = 9 \)[/tex]
- [tex]\( 9 \div 3 = 3 \)[/tex]
- [tex]\( 3 \div 3 = 1 \)[/tex]
So, [tex]\( 1728 = 2^6 \times 3^3 \)[/tex].
#### c. 5000
For 5000, we break it down as follows:
- [tex]\( 5000 \div 2 = 2500 \)[/tex]
- [tex]\( 2500 \div 2 = 1250 \)[/tex]
- [tex]\( 1250 \div 2 = 625 \)[/tex]
- [tex]\( 625 \div 5 = 125 \)[/tex]
- [tex]\( 125 \div 5 = 25 \)[/tex]
- [tex]\( 25 \div 5 = 5 \)[/tex]
- [tex]\( 5 \div 5 = 1 \)[/tex]
So, [tex]\( 5000 = 2^3 \times 5^4 \)[/tex].
#### d. 548
For 548, we break it down as follows:
- [tex]\( 548 \div 2 = 274 \)[/tex]
- [tex]\( 274 \div 2 = 137 \)[/tex]
- Since 137 is a prime number, it is not divisible further by any primes other than itself.
So, [tex]\( 548 = 2^2 \times 137 \)[/tex].
### Summary
To summarize:
- For [tex]\( 6 \)[/tex]:
- a. [tex]\( 2^5 = 32 \)[/tex] is bigger.
- b. [tex]\( 3^4 = 81 \)[/tex] is bigger.
- For [tex]\( 7 \)[/tex]:
- a. [tex]\( 324 = 2^2 \times 3^4 \)[/tex]
- b. [tex]\( 1728 = 2^6 \times 3^3 \)[/tex]
- c. [tex]\( 5000 = 2^3 \times 5^4 \)[/tex]
- d. [tex]\( 548 = 2^2 \times 137 \)[/tex]
