Answer :
Sure, let's break this down step-by-step:
### Part (a): Express 3125 as a product of prime factors in exponential form
We start by finding the prime factors of 3125.
1. Notice that 3125 ends in 5, suggesting it is divisible by 5.
2. Divide 3125 by 5:
[tex]\[ 3125 \div 5 = 625 \][/tex]
3. Repeat the process with 625:
[tex]\[ 625 \div 5 = 125 \][/tex]
4. Continue with 125:
[tex]\[ 125 \div 5 = 25 \][/tex]
5. Next, divide 25:
[tex]\[ 25 \div 5 = 5 \][/tex]
6. Finally, divide 5:
[tex]\[ 5 \div 5 = 1 \][/tex]
So, we have:
[tex]\[ 3125 = 5 \times 5 \times 5 \times 5 \times 5 \][/tex]
This can be expressed in exponential form as:
[tex]\[ 3125 = 5^5 \][/tex]
### Part (b): Express [tex]\( 729 \times 128 \)[/tex] as a product of prime factors in exponential form
We first factorize 729 and 128 individually into their prime factors.
#### Step 1: Factorize 729
1. Notice that 729 ends in an odd digit, suggesting checking for smaller prime factors.
2. It turns out 729 is a power of 3. We can find successively:
[tex]\[ 729 \div 3 = 243 \][/tex]
[tex]\[ 243 \div 3 = 81 \][/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, we have:
[tex]\[ 729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \][/tex]
This can be expressed in exponential form as:
[tex]\[ 729 = 3^6 \][/tex]
#### Step 2: Factorize 128
1. Notice that 128 is even, suggesting divisibility by 2:
2. Divide 128 by 2 repeatedly:
[tex]\[ 128 \div 2 = 64 \][/tex]
[tex]\[ 64 \div 2 = 32 \][/tex]
[tex]\[ 32 \div 2 = 16 \][/tex]
[tex]\[ 16 \div 2 = 8 \][/tex]
[tex]\[ 8 \div 2 = 4 \][/tex]
[tex]\[ 4 \div 2 = 2 \][/tex]
[tex]\[ 2 \div 2 = 1 \][/tex]
So, we have:
[tex]\[ 128 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \][/tex]
This can be expressed in exponential form as:
[tex]\[ 128 = 2^7 \][/tex]
#### Step 3: Combine the prime factors for [tex]\( 729 \times 128 \)[/tex]
Now we multiply the prime factors of 729 and 128:
[tex]\[ 729 \times 128 = 3^6 \times 2^7 \][/tex]
So, the expression for [tex]\( 729 \times 128 \)[/tex] in prime factorized exponential form is:
[tex]\[ 729 \times 128 = 2^7 \times 3^6 \][/tex]
### Summary
(a) [tex]\( 3125 = 5^5 \)[/tex]
(b) [tex]\( 729 \times 128 = 2^7 \times 3^6 \)[/tex]
### Part (a): Express 3125 as a product of prime factors in exponential form
We start by finding the prime factors of 3125.
1. Notice that 3125 ends in 5, suggesting it is divisible by 5.
2. Divide 3125 by 5:
[tex]\[ 3125 \div 5 = 625 \][/tex]
3. Repeat the process with 625:
[tex]\[ 625 \div 5 = 125 \][/tex]
4. Continue with 125:
[tex]\[ 125 \div 5 = 25 \][/tex]
5. Next, divide 25:
[tex]\[ 25 \div 5 = 5 \][/tex]
6. Finally, divide 5:
[tex]\[ 5 \div 5 = 1 \][/tex]
So, we have:
[tex]\[ 3125 = 5 \times 5 \times 5 \times 5 \times 5 \][/tex]
This can be expressed in exponential form as:
[tex]\[ 3125 = 5^5 \][/tex]
### Part (b): Express [tex]\( 729 \times 128 \)[/tex] as a product of prime factors in exponential form
We first factorize 729 and 128 individually into their prime factors.
#### Step 1: Factorize 729
1. Notice that 729 ends in an odd digit, suggesting checking for smaller prime factors.
2. It turns out 729 is a power of 3. We can find successively:
[tex]\[ 729 \div 3 = 243 \][/tex]
[tex]\[ 243 \div 3 = 81 \][/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, we have:
[tex]\[ 729 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \][/tex]
This can be expressed in exponential form as:
[tex]\[ 729 = 3^6 \][/tex]
#### Step 2: Factorize 128
1. Notice that 128 is even, suggesting divisibility by 2:
2. Divide 128 by 2 repeatedly:
[tex]\[ 128 \div 2 = 64 \][/tex]
[tex]\[ 64 \div 2 = 32 \][/tex]
[tex]\[ 32 \div 2 = 16 \][/tex]
[tex]\[ 16 \div 2 = 8 \][/tex]
[tex]\[ 8 \div 2 = 4 \][/tex]
[tex]\[ 4 \div 2 = 2 \][/tex]
[tex]\[ 2 \div 2 = 1 \][/tex]
So, we have:
[tex]\[ 128 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \][/tex]
This can be expressed in exponential form as:
[tex]\[ 128 = 2^7 \][/tex]
#### Step 3: Combine the prime factors for [tex]\( 729 \times 128 \)[/tex]
Now we multiply the prime factors of 729 and 128:
[tex]\[ 729 \times 128 = 3^6 \times 2^7 \][/tex]
So, the expression for [tex]\( 729 \times 128 \)[/tex] in prime factorized exponential form is:
[tex]\[ 729 \times 128 = 2^7 \times 3^6 \][/tex]
### Summary
(a) [tex]\( 3125 = 5^5 \)[/tex]
(b) [tex]\( 729 \times 128 = 2^7 \times 3^6 \)[/tex]
