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The square root property is a powerful tool when solving quadratic equations. It tells us that if you have an equation in the form of \( x^2 = k \), then the solutions are \( x = \pm \sqrt{k} \). This property is extremely useful because it allows us to solve for \( x \) by taking the square root of both sides.
In the example given:
\[ 7x^{2} = 4\]
After isolating \( x^2 \) by dividing both sides by 7, we get:
\[ x^{2} = \frac{4}{7}\]
Applying the square root property here, we take the square root of both sides:
\[ x = \pm \sqrt{\frac{4}{7}} \]
Don't forget the \( \pm \) sign because squaring either a positive or a negative number yields a positive result. This method applies broadly to any quadratic equation that can be manipulated into the form \( x^2 = k \).

To make the square root expression simpler, we apply the technique of radical simplification. This technique helps us to break down square root expressions into simpler, more recognizable terms.
Starting from:
\[ x = \pm \sqrt{\frac{4}{7}} \]
We use the property that the square root of a fraction equals the square root of its numerator divided by the square root of its denominator:
\[ x = \pm \frac{\sqrt{4}}{\sqrt{7}} \]
Here, \( \sqrt{4} \) simplifies to 2, so we have:
\[ x = \pm \frac{2}{\sqrt{7}} \]
This technique of simplifying the radicals helps in dealing with root expressions more effectively in various mathematical problems.

Rationalizing the denominator is a useful technique to remove any radicals from the denominators of fractions. A fraction is easier to work with if the denominator is a rational number.
In our problem, we have:
\[ x = \pm \frac{2}{\sqrt{7}} \]
To rationalize this, we multiply both the numerator and the denominator by \( \sqrt{7} \):
\[ x = \pm \frac{2 \sqrt{7}}{\sqrt{7} \sqrt{7}} = \pm \frac{2 \sqrt{7}}{7} \]
Now, the denominator is rational (since \( \sqrt{7} \sqrt{7} = 7 \)), and our expression simplifies to:
\[ x = \pm \frac{2 \sqrt{7}}{7} \]
This process of rationalizing the denominator makes the expression clearer and easier to handle in further mathematical operations.