21 \( \sqrt{1-\frac{7}{8}}-\frac{8}{9} \cdot\left(1-\frac{1}{4}\right)^{3} \) \( 1-7=\sqrt{6}=8-8 \)

Let's simplify the expression step by step. The expression is: \[ 21 \sqrt{1 - \frac{7}{8}} - \frac{8}{9} \cdot \left(1 - \frac{1}{4}\right)^{3} \] ### Step 1: Simplify \( \sqrt{1 - \frac{7}{8}} \) First, calculate \( 1 - \frac{7}{8} \): \[ 1 - \frac{7}{8} = \frac{8}{8} - \frac{7}{8} = \frac{1}{8} \] Now, take the square root: \[ \sqrt{1 - \frac{7}{8}} = \sqrt{\frac{1}{8}} = \frac{1}{\sqrt{8}} = \frac{1}{2\sqrt{2}} = \frac{\sqrt{2}}{4} \] ### Step 2: Calculate \( 21 \sqrt{1 - \frac{7}{8}} \) Now, multiply by 21: \[ 21 \sqrt{1 - \frac{7}{8}} = 21 \cdot \frac{\sqrt{2}}{4} = \frac{21\sqrt{2}}{4} \] ### Step 3: Simplify \( \left(1 - \frac{1}{4}\right)^{3} \) Calculate \( 1 - \frac{1}{4} \): \[ 1 - \frac{1}{4} = \frac{3}{4} \] Now, raise it to the power of 3: \[ \left(\frac{3}{4}\right)^{3} = \frac{27}{64} \] ### Step 4: Calculate \( \frac{8}{9} \cdot \left(1 - \frac{1}{4}\right)^{3} \) Now, multiply by \( \frac{8}{9} \): \[ \frac{8}{9} \cdot \frac{27}{64} = \frac{8 \cdot 27}{9 \cdot 64} = \frac{216}{576} \] Now simplify \( \frac{216}{576} \): \[ \frac{216 \div 72}{576 \div 72} = \frac{3}{8} \] ### Step 5: Combine the results Now we have: \[ 21 \sqrt{1 - \frac{7}{8}} - \frac{8}{9} \cdot \left(1 - \frac{1}{4}\right)^{3} = \frac{21\sqrt{2}}{4} - \frac{3}{8} \] To combine these fractions, we need a common denominator. The common denominator of 4 and 8 is 8. Convert \( \frac{21\sqrt{2}}{4} \) to have a denominator of 8: \[ \frac{21\sqrt{2}}{4} = \frac{21\sqrt{2} \cdot 2}{4 \cdot 2} = \frac{42\sqrt{2}}{8} \] Now we can combine: \[ \frac{42\sqrt{2}}{8} - \frac{3}{8} = \frac{42\sqrt{2} - 3}{8} \] ### Final Result Thus, the final simplified expression is: \[ \frac{42\sqrt{2} - 3}{8} \]