Simplify square root of square root of square root of 256/6561

$\sqrt{\sqrt{\sqrt{\frac{256}{6561}}}}$

Step 1

Rewrite $\sqrt{\frac{256}{6561}}$ as $\frac{\sqrt{256}}{\sqrt{6561}}$ .

$\sqrt{\sqrt{\frac{\sqrt{256}}{\sqrt{6561}}}}$

Step 2

Simplify the numerator.

Rewrite $256$ as $16^{2}$ .

$\sqrt{\sqrt{\frac{\sqrt{16^{2}}}{\sqrt{6561}}}}$

Pull terms out from under the radical, assuming positive real numbers.

$\sqrt{\sqrt{\frac{16}{\sqrt{6561}}}}$

Step 3

Simplify the denominator.

Rewrite $6561$ as $81^{2}$ .

$\sqrt{\sqrt{\frac{16}{\sqrt{81^{2}}}}}$

Pull terms out from under the radical, assuming positive real numbers.

$\sqrt{\sqrt{\frac{16}{81}}}$

Step 4

Rewrite $\sqrt{\frac{16}{81}}$ as $\frac{\sqrt{16}}{\sqrt{81}}$ .

$\sqrt{\frac{\sqrt{16}}{\sqrt{81}}}$

Step 5

Simplify the numerator.

Rewrite $16$ as $4^{2}$ .

$\sqrt{\frac{\sqrt{4^{2}}}{\sqrt{81}}}$

Pull terms out from under the radical, assuming positive real numbers.

$\sqrt{\frac{4}{\sqrt{81}}}$

Step 6

Simplify the denominator.

Rewrite $81$ as $9^{2}$ .

$\sqrt{\frac{4}{\sqrt{9^{2}}}}$

Pull terms out from under the radical, assuming positive real numbers.

$\sqrt{\frac{4}{9}}$

Step 7

Rewrite $\sqrt{\frac{4}{9}}$ as $\frac{\sqrt{4}}{\sqrt{9}}$ .

$\frac{\sqrt{4}}{\sqrt{9}}$

Step 8

Simplify the numerator.

Rewrite $4$ as $2^{2}$ .

$\frac{\sqrt{2^{2}}}{\sqrt{9}}$

Pull terms out from under the radical, assuming positive real numbers.

$\frac{2}{\sqrt{9}}$

Step 9

Simplify the denominator.

Rewrite $9$ as $3^{2}$ .

$\frac{2}{\sqrt{3^{2}}}$

Pull terms out from under the radical, assuming positive real numbers.

$\frac{2}{3}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{3}$

Decimal Form:

$0.\overline{6}$

Simplify square root of square root of square root of 256/6561

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