Questions: Simplify. (3 - √72) / 9

Solution Steps

To simplify the expression \(\frac{3-\sqrt{72}}{9}\), we need to first simplify the square root in the numerator. The square root of 72 can be broken down into its prime factors, and then simplified. After simplifying the square root, we can simplify the entire fraction by dividing both the numerator and the denominator by their greatest common divisor.

First, we need to simplify the square root in the expression \(\sqrt{72}\). We can do this by finding the prime factorization of 72:

\[ 72 = 2^3 \times 3^2 \]

Using the property of square roots, \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\), we can simplify:

\[ \sqrt{72} = \sqrt{2^3 \times 3^2} = \sqrt{2^2 \times 2 \times 3^2} = \sqrt{2^2} \times \sqrt{3^2} \times \sqrt{2} = 2 \times 3 \times \sqrt{2} = 6\sqrt{2} \]

Substitute \(\sqrt{72} = 6\sqrt{2}\) back into the original expression:

\[ \frac{3 - \sqrt{72}}{9} = \frac{3 - 6\sqrt{2}}{9} \]

Now, simplify the fraction by dividing each term in the numerator by the denominator:

\[ \frac{3 - 6\sqrt{2}}{9} = \frac{3}{9} - \frac{6\sqrt{2}}{9} \]

Simplify each term:

\[ \frac{3}{9} = \frac{1}{3} \]

\[ \frac{6\sqrt{2}}{9} = \frac{2\sqrt{2}}{3} \]

Thus, the expression becomes:

\[ \frac{1}{3} - \frac{2\sqrt{2}}{3} \]

Combine the terms:

\[ \frac{1 - 2\sqrt{2}}{3} \]

Final Answer