Answer:
9z
Step-by-step explanation:
4th root of z = z^1/4
=> 3 * z^1/4
=> 3z^1/4
3z^1/4 * 3z^3/4
=> 3 x 3 x z^1/4 + 3/4
=> 9z^4/4
=> 9z^1
=> 9z
The expression [tex]3\sqrt[4]{z}~.~3z^{\frac{3}{4} }[/tex] in the form k . [tex]z^{n}[/tex] is 9z.
We are given:
= [tex]3\sqrt[4]{z}~.~3z^{\frac{3}{4} }[/tex]
We need to write this in the form of :
= [tex]k~.~z^{n}[/tex]
We use exponents and powers when we want to write very large numbers or very small numbers in a simplified manner.
Example:
8 = 2 x 2 x 2 = [tex]2^{3}[/tex].
Where 2 is the base and 3 is the exponents.
72 = 8 x 9 = [tex]2^{3} \times3^{2}[/tex].
Here we have two bases 2 and 3 and two exponents 3 and 2.
8 / 9 = [tex]\frac{2^{3} }{3^{2} }[/tex] = [tex]2^{3} \times3^{-2}[/tex].
Here the bases are 2 and 3 and exponents are 3 and -2.
We have
[tex]3\sqrt[4]{z}~.~3z^{\frac{3}{4} }[/tex]
Here,
[tex]\sqrt[4]{z}[/tex] can be written as [tex]z^{\frac{1}{4} }[/tex].
[tex]z^{\frac{1}{4} }~.~z^{\frac{3}{4} }[/tex] can be written as :
[tex]z^{\frac{1}{4} +\frac{3}{4} }\\\\ z^{\frac{4}{4} } \\\\z^{1} = z[/tex]
Now we have,
3 x 3 x [tex]z^{\frac{1}{4} }[/tex] x [tex]z^{\frac{3}{4} }[/tex]
3 x 3 x z
[tex]3^{2}[/tex] x z
9 x [tex]z^{1}[/tex]
So, comparing with k . [tex]z^{n}[/tex]
we have,
k = 9 and n = 1.
The expression [tex]3\sqrt[4]{z}~.~3z^{\frac{3}{4} }[/tex] in the form k . [tex]z^{n}[/tex] is 9z.
Learn more about exponents and powers here:
https://brainly.com/question/9648446
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