Apply the square root property sqrt((a)/(b)) = sqrt(a)sqrt(b).

Here are the solutions to the problems:

: Simplify $\sqrt{\frac{32}{81}}$

Step 1: Apply the square root property $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$. $\sqrt{\frac{32}{81}} = \frac{\sqrt{32}}{\sqrt{81}}$

Step 2: Simplify the square roots in the numerator and denominator. We know that $\sqrt{81} = 9$. For $\sqrt{32}$, find the largest perfect square factor of 32. $32 = 16 \times 2$. $\sqrt{32} = \sqrt{16 \times 2} = \sqrt{16} \times \sqrt{2} = 4\sqrt{2}$

Step 3: Substitute the simplified square roots back into the fraction. $\frac{4\sqrt{2}}{9}$ The simplified expression is: \frac{4\sqrt{2}{9}}

: Evaluate $\sqrt{\frac{32}{128}} - \frac{1}{3}$

Step 1: Simplify the fraction inside the square root. $\frac{32}{128} = \frac{32 \div 32}{128 \div 32} = \frac{1}{4}$

Step 2: Calculate the square root of the simplified fraction. $\sqrt{\frac{1}{4}} = \frac{\sqrt{1}}{\sqrt{4}} = \frac{1}{2}$

Step 3: Perform the subtraction. $\frac{1}{2} - \frac{1}{3}$ To subtract, find a common denominator, which is 6. $\frac{1 \times 3}{2 \times 3} - \frac{1 \times 2}{3 \times 2} = \frac{3}{6} - \frac{2}{6}$

Step 4: Subtract the numerators. $\frac{3 - 2}{6} = \frac{1}{6}$ The result is: \frac{1{6}}

: Evaluate $0.13 \times 0.05$

Step 1: Multiply the numbers as if they were whole numbers. $13 \times 5 = 65$

Step 2: Count the total number of decimal places in the original numbers. $0.13$ has 2 decimal places. $0.05$ has 2 decimal places. Total decimal places = $2 + 2 = 4$.

Step 3: Place the decimal point in the product so that it has 4 decimal places. Starting with 65, move the decimal point 4 places to the left, adding leading zeros as needed. $0.0065$ The product is: $0.0065$

Send me the next one 📸