We need to find all the real fourth roots of 2562401.

Let us write,x=256240114⇒x4=2562401On factorization of the given fraction, we can write2562401=16492=474Therefore,x4=474⇒x4-474=0⇒x2-472x2+472=0∵a2-b2=a+ba-b⇒x-47x+47x2+1649=0⇒x=47,-47Hence, the real fourth roots of 2562401 is 47&amp;-47.

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Algebra
Pre-Algebra
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kapitlio0z

Answered question

2022-12-16

We need to find all the real fourth roots of

\frac{256}{2401}

Answer & Explanation

rerudindu9z3

Beginner2022-12-17Added 2 answers

Let us write,

x = {(\frac{256}{2401})}^{\frac{1}{4}} \Rightarrow x^{4} = \frac{256}{2401}

On factorization of the given fraction, we can write

\frac{256}{2401} = {(\frac{16}{49})}^{2} = {(\frac{4}{7})}^{4}

Therefore,

x^{4} = {(\frac{4}{7})}^{4} \Rightarrow x^{4} - {(\frac{4}{7})}^{4} = 0 \Rightarrow (x^{2} - {(\frac{4}{7})}^{2}) (x^{2} + {(\frac{4}{7})}^{2}) = 0 (∵ (a^{2} - b^{2}) = (a + b) (a - b)) \Rightarrow (x - \frac{4}{7}) (x + \frac{4}{7}) (x^{2} + \frac{16}{49}) = 0 \Rightarrow x = \frac{4}{7}, - \frac{4}{7}

Hence, the real fourth roots of

\frac{256}{2401}

\frac{4}{7} & - \frac{4}{7}

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