Sure, let’s write each of these products using index notation and simplify the expressions step by step.
### a) [tex]\( 4 \times 4 \times 4 \times 4 \times 4 \times 4 \)[/tex]
In index notation:
[tex]\[ 4 \times 4 \times 4 \times 4 \times 4 \times 4 = 4^6 \][/tex]
Calculating [tex]\( 4^6 \)[/tex]:
[tex]\[ 4^6 = 4096 \][/tex]
### b) [tex]\( 5 \times 5 \times 5 \)[/tex]
In index notation:
[tex]\[ 5 \times 5 \times 5 = 5^3 \][/tex]
Calculating [tex]\( 5^3 \)[/tex]:
[tex]\[ 5^3 = 125 \][/tex]
### c) [tex]\( 3 \times 3 \times 3 \times 3 \times 7 \)[/tex]
In index notation:
[tex]\[ 3 \times 3 \times 3 \times 3 \times 7 = 3^4 \times 7 \][/tex]
Calculating [tex]\( 3^4 \)[/tex]:
[tex]\[ 3^4 = 81 \][/tex]
Now multiplying by 7:
[tex]\[ 81 \times 7 = 567 \][/tex]
### d) [tex]\( 2 \times 9 \times 2 \times 9 \times 9 \)[/tex]
In index notation:
[tex]\[ 2 \times 9 \times 2 \times 9 \times 9 = (2 \times 2) \times (9 \times 9 \times 9) \][/tex]
Rewrite as:
[tex]\[ 2^2 \times 9^3 \][/tex]
Calculating [tex]\( 2^2 \)[/tex]:
[tex]\[ 2^2 = 4 \][/tex]
Calculating [tex]\( 9^3 \)[/tex]:
[tex]\[ 9^3 = 729 \][/tex]
Now multiplying 4 and 729:
[tex]\[ 4 \times 729 = 2916 \][/tex]
### Summary
- a) [tex]\( 4^6 = 4096 \)[/tex]
- b) [tex]\( 5^3 = 125 \)[/tex]
- c) [tex]\( 3^4 \times 7 = 567 \)[/tex]
- d) [tex]\( 2^2 \times 9^3 = 2916 \)[/tex]
These calculations yield the final results as follows:
- [tex]\( 4^6 = 4096 \)[/tex]
- [tex]\( 5^3 = 125 \)[/tex]
- [tex]\( 3^4 \times 7 = 567 \)[/tex]
- [tex]\( 2^2 \times 9^3 = 2916 \)[/tex]
