Respuesta :
32. [tex]256^{\frac{1}{2} }[/tex] can be simplified into 512.
Prime factorization is the process of breaking down a number into its prime factors. Multiplying these prime numbers gives back the original number.
Prime factorization of 32 = 2 x 2 x 2 x 2 x 2 = [tex]2^{5}[/tex]
Prime factorization of 256= 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = [tex]2^{8}[/tex]
According to the law of indices, if a term with a power is raised to a power, then the powers are multiplied together.
i.e., [tex](x^{m})^{n} = x^{mn}[/tex]
Another law of indices says that if two terms having same base are multiplied together, then their indices are added.
i.e., [tex]a^{m} . a^{n } =a^{m+n}[/tex]
According to the given condition,
∴[tex]32 = 2^{5}[/tex]
[tex]256^{\frac{1}{2}} = (2^{8})^{\frac{1}{2} }[/tex]
∴ 32. [tex]256^{\frac{1}{2} }[/tex] = [tex]2^{5}[/tex] × [tex](2^{8})^{\frac{1}{2} }[/tex]
Applying the laws of indices mentioned above,
32. [tex]256^{\frac{1}{2} }[/tex] = [tex]2^{5}[/tex] × [tex](2^{8 X }^{\frac{1}{2} })[/tex]
= [tex]2^{5}[/tex] × [tex]2^{4}[/tex]
= [tex]2^{5 + 4}[/tex] = [tex]2^{9}[/tex]
= 512.
Thus, 32. [tex]256^{\frac{1}{2} }[/tex] can be simplified into 512.
To learn more about laws of indices, refer to this link:
brainly.com/question/27432311
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