Answer:
Real roots of [tex]\frac{256}{2401}[/tex] are [tex]{\frac{4}{7}}[/tex] and [tex]{\frac{-4}{7}}[/tex]
Step-by-step explanation:
We have to find the fourth root of [tex]\frac{256}{2401}[/tex]
that is [tex]x^4=\frac{256}{2401}[/tex]
[tex]\Rightarrow x^4-\frac{256}{2401}=0[/tex]
Using identity , [tex]a^2-b^2= (a+b)(a-b)[/tex]
[tex]\Rightarrow (x^2-\frac{16}{49})(x^2+\frac{16}{49})=0[/tex]
Using zero product rule, it states if product of two number is zero then either first number is zero or second number is zero that is, [tex]a.b=0 \Rightarrow a=0 \ \text{or} \ b=0[/tex]
thus,
[tex]\Rightarrow (x^2-\frac{16}{49})=0[/tex] or [tex]\Rightarrow (x^2+\frac{16}{49})=0[/tex]
[tex]\Rightarrow x^2=\frac{16}{49}[/tex] or [tex]\Rightarrow x^2=\frac{-16}{49}[/tex]
[tex]\Rightarrow x^2=\frac{-16}{49}[/tex] this will complex roots.
[tex]\Rightarrow x^2=\frac{16}{49}[/tex]
[tex]\Rightarrow x=\sqrt{\frac{16}{49}}[/tex]
[tex]\Rightarrow x=\pm{\frac{4}{7}}[/tex]
Thus, real roots of [tex]\frac{256}{2401}[/tex] are [tex]{\frac{4}{7}}[/tex] and [tex]{\frac{-4}{7}}[/tex]
