Simplify square root of 7/(9a^2)

Solution:

Step 1:

Express $\frac{7}{9a^2}$ in the form of ${\left(\frac{1}{3a}\right)}^2\cdot 7$.

Step 1.1:

Extract the square of $1$ from the numerator in $7$ as $\sqrt{\frac{1^2\cdot 7}{9a^2}}$.

Step 1.2:

Extract the square of $3a$ from the denominator in $9a^2$ as $\sqrt{\frac{1^2\cdot 7}{(3a{)}^2\cdot 1}}$.

Step 1.3:

Reorganize the fraction $\frac{1^2\cdot 7}{(3a{)}^2\cdot 1}$ to $\sqrt{{\left(\frac{1}{3a}\right)}^2\cdot 7}$.

Step 2:

Remove terms from under the square root to get $\frac{1}{3a}\sqrt{7}$.

Step 3:

Merge the terms $\frac{1}{3a}$ and $\sqrt{7}$ to obtain the simplified form $\frac{\sqrt{7}}{3a}$.

Knowledge Notes:

To simplify the square root of a fraction, we can apply the following knowledge points:

1. Square Root of a Fraction: The square root of a fraction can be simplified by taking the square root of the numerator and the denominator separately. If $\sqrt{\frac{a}{b}}$ is given, it can be simplified to $\frac{\sqrt{a}}{\sqrt{b}}$.

2. Simplifying Square Roots: To simplify square roots, we look for perfect squares in the radicand (the number under the square root). If a perfect square can be factored out, it can be taken out of the square root as its base.

3. Rationalizing the Denominator: When we have a square root in the denominator, we often rationalize it by multiplying the numerator and the denominator by a suitable expression that will eliminate the square root in the denominator.

4. Algebraic Manipulation: Algebraic expressions can often be rewritten in different forms to simplify calculations. In this case, we rewrite $\frac{7}{9a^2}$ as ${\left(\frac{1}{3a}\right)}^2\cdot 7$ to make it easier to take the square root.

5. Perfect Squares: Recognizing perfect squares is essential in simplifying square roots. For example, $1^2=1$ and $(3a{)}^2=9a^2$ are perfect squares that can be used to simplify the given expression.

In the provided solution, we used these principles to simplify the square root of the fraction by expressing the denominator as a square of a simpler expression, extracting perfect squares, and then combining the terms to get the final simplified form.