[tex]5N^2=2^2 \times 5^{7} \times x^{8}[/tex]
Solution:
Given a number N in product of its prime factors.
[tex]N=2 \times 5^{3} \times x^{4}[/tex]
Now, express [tex]5 N^{2}[/tex] as a product of prime factors in index form.
Substitute [tex]2 \times 5^{3} \times x^{4}[/tex] in place of N.
[tex]5N^2=5\times(2 \times 5^{3} \times x^{4})^2[/tex]
Use exponential rule: [tex]\left(a^{m}\right)^{n}=a^{m n}[/tex]
Multiply the power 2 into the powers of factors of every number in the bracket.
[tex]5N^2=5\times2^2 \times 5^{6} \times x^{8}[/tex]
Arrange like terms together.
[tex]5N^2=2^2 \times 5\times 5^{6} \times x^{8}[/tex]
Use exponential rule: [tex]a^{m} \times a^{n}=a^{m+n}[/tex]
[tex]5N^2=2^2 \times 5^{1+6} \times x^{8}[/tex]
[tex]5N^2=2^2 \times 5^{7} \times x^{8}[/tex]
Hence [tex]5N^2=2^2 \times 5^{7} \times x^{8}[/tex].
