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Rewrite the following using index notation:

a) [tex]\(4 \times 4 \times 4 \times 4 \times 4 \times 4\)[/tex]

b) [tex]\(5 \times 5 \times 5\)[/tex]

c) [tex]\(3 \times 3 \times 3 \times 3 \times 7\)[/tex]

d) [tex]\(2 \times 9 \times 2 \times 9 \times 9\)[/tex]

Sagot :

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]