The index notation involves writing numbers in index form.
The index forms are:[tex]\mathbf{3 \times 7^{4}}[/tex] and [tex]\mathbf{2^{2} \times 9^{4}}[/tex]
[tex]\mathbf{(a)\ 3 \times 7 \times 7 \times 7 \times 7}[/tex]
Rewrite as:
[tex]\mathbf{3 \times 7 \times 7 \times 7 \times 7 = 3 \times 7^1 \times 7^1 \times 7^1 \times 7^1}[/tex]
Apply law of indices
[tex]\mathbf{3 \times 7 \times 7 \times 7 \times 7 = 3 \times 7^{1+1+1+1}}[/tex]
[tex]\mathbf{3 \times 7 \times 7 \times 7 \times 7 = 3 \times 7^{4}}[/tex]
Hence, the index form is: [tex]\mathbf{3 \times 7^{4}}[/tex]
[tex]\mathbf{(b)\ 2 \times 9 \times 9 \times 9 \times 2 \times 9}[/tex]
Rewrite as:
[tex]\mathbf{2 \times 9 \times 9 \times 9 \times 2 \times 9 = 2^1 \times 9^1 \times 9^1 \times 9^1 \times 2^1 \times 9^1}[/tex]
Rewrite as:
[tex]\mathbf{2 \times 9 \times 9 \times 9 \times 2 \times 9 = 2^1\times 2^1 \times 9^1 \times 9^1 \times 9^1 \times 9^1}[/tex]
Apply law of indices
[tex]\mathbf{2 \times 9 \times 9 \times 9 \times 2 \times 9 = 2^{1+1} \times 9^{1+1+1+1}}[/tex]
[tex]\mathbf{2 \times 9 \times 9 \times 9 \times 2 \times 9 = 2^{2} \times 9^{4}}[/tex]
Hence, the index form is: [tex]\mathbf{2^{2} \times 9^{4}}[/tex]
Read more about the index form of expressions at:
https://brainly.com/question/23340352
