Understanding how to simplify radicals, especially when dealing with fractions, is essential in algebra. The key principle is that if a term can be expressed as a product, it can be separated into individual radicals. This technique can also be applied to fractions, allowing us to simplify complex expressions more easily.

When simplifying a radical that contains a fraction, we can break it down into the square root of the numerator and the square root of the denominator. For example, consider the fraction \(\frac{49}{64}\). Both 49 and 64 are perfect squares, so we can express this as \(\sqrt{49} / \sqrt{64}\), which simplifies to \(\frac{7}{8}\).

In another example, simplifying \(\frac{\sqrt{32}}{\sqrt{2}}\) can be approached by combining the radicals into one: \(\sqrt{\frac{32}{2}} = \sqrt{16}\), which simplifies to 4, a perfect square.

When variables are involved, the same principles apply. For instance, consider \(\sqrt{\frac{64x^4}{9x^2}}\). Instead of trying to simplify the fraction directly, we can separate it into \(\frac{\sqrt{64x^4}}{\sqrt{9x^2}}\). Here, \(\sqrt{64} = 8\) and \(\sqrt{x^4} = x^2\), while \(\sqrt{9} = 3\) and \(\sqrt{x^2} = x\). This results in \(\frac{8x^2}{3x}\), which can be further simplified to \(\frac{8}{3}x\) by applying the quotient rule for exponents.

In another scenario, simplifying \(\sqrt{\frac{72x^3}{9x}}\) involves first simplifying the fraction to \(\sqrt{8x^2}\). Recognizing that 8 can be expressed as \(\sqrt{4 \cdot 2}\) and \(\sqrt{x^2} = x\), we can rewrite this as \(2x\sqrt{2}\), which is fully simplified.

In summary, the process of simplifying radicals, especially those involving fractions and variables, relies on recognizing perfect squares and applying the properties of exponents. By breaking down complex expressions into manageable parts, we can achieve a clearer and more simplified result.