Sure, let's proceed to rewrite each expression using index notation step-by-step:
### a) [tex]\(2 \times 2 \times 2\)[/tex]
Step-by-Step:
1. Count how many times the number 2 appears in the multiplication.
- There are three 2s.
2. When you multiply a number by itself multiple times, it is expressed as the base raised to the power of the number of times it is multiplied.
Index Notation:
[tex]\[2 \times 2 \times 2 = 2^3\][/tex]
### b) [tex]\(3 \times 3 \times 3 \times 3 \times 3\)[/tex]
Step-by-Step:
1. Count how many times the number 3 appears in the multiplication.
- There are five 3s.
2. Express this product using exponentiation.
Index Notation:
[tex]\[3 \times 3 \times 3 \times 3 \times 3 = 3^5\][/tex]
### c) [tex]\(4 \times 4 \times 4 \times 5 \times 5\)[/tex]
Step-by-Step:
1. Separate the product into two groups: one containing all the 4s and the other containing all the 5s.
- The first group contains three 4s.
- The second group contains two 5s.
2. Express each group using exponentiation.
Index Notation:
[tex]\[4 \times 4 \times 4 \times 5 \times 5 = 4^3 \times 5^2\][/tex]
### d) [tex]\(9 \times 7 \times 9 \times 9 \times 7 \times 9\)[/tex]
Step-by-Step:
1. Group the same numbers together: the 9s and the 7s.
- The first group contains four 9s.
- The second group contains two 7s.
2. Express each group using exponentiation.
Index Notation:
[tex]\[9 \times 7 \times 9 \times 9 \times 7 \times 9 = 9^4 \times 7^2\][/tex]
Thus, the given multiplications simplified using index notation are:
a) [tex]\(2 \times 2 \times 2 = 2^3\)[/tex]
b) [tex]\(3 \times 3 \times 3 \times 3 \times 3 = 3^5\)[/tex]
c) [tex]\(4 \times 4 \times 4 \times 5 \times 5 = 4^3 \times 5^2\)[/tex]
d) [tex]\(9 \times 7 \times 9 \times 9 \times 7 \times 9 = 9^4 \times 7^2\)[/tex]
