Express $$ \frac{5}{\sqrt{8 n^{6}}} $$ in simplest radical form with a rational denominator (assume $$ n0 $$ ).

Question

Express $$ \frac{5}{\sqrt{8 n^{6}}} $$ in simplest radical form with a rational denominator (assume $$ n>0 $$ ).

UpStudy AI · Accepted Answer

We start with the expression

5/√(8n⁶).

Step 1. Rewrite the square root by factoring:
  √(8n⁶) = √8 · √(n⁶).

Notice that
  √8 = 2√2   and  √(n⁶) = n³
(since n > 0, we take the positive cube).

Thus,
  √(8n⁶) = 2n³√2.

Step 2. Substitute back into the original expression:
  5/√(8n⁶) = 5/(2n³√2).

Step 3. Rationalize the denominator by multiplying the numerator and denominator by √2:
  5/(2n³√2) = (5√2)/(2n³·2) = (5√2)/(4n³).

The expression in its simplest radical form with a rational denominator is

(5√2)/(4n³).
The simplified form of $$ \frac{5}{\sqrt{8n^{6}}} $$ with a rational denominator is $$ \frac{5\sqrt{2}}{4n^{3}} $$.

Express in simplest radical form with a rational denominator (assume ).