Question

Challenge Problems. Perform the indicated operation and simplify. $$\sqrt{\frac{9 x}{16}}-\frac{3 \sqrt{x}}{4}$$

Short answer

The simplified result is \(0\).

Step-by-step solution

Simplify the square root

Begin by simplifying the square root of the fraction. The square root of a fraction \(\sqrt{\frac{a}{b}}\) is equal to the square root of the numerator divided by the square root of the denominator. So \(\sqrt{\frac{9x}{16}}\) simplifies to \(\frac{\sqrt{9x}}{\sqrt{16}}\), which is \(\frac{3\sqrt{x}}{4}\).

Rewrite the expression with simplified square root

Rewrite the original expression using the simplified form of the square root: \(\frac{3\sqrt{x}}{4} - \frac{3\sqrt{x}}{4}\).

Combine like terms

Since both terms are now exactly the same, with both having \(\frac{3\sqrt{x}}{4}\) but with different signs, they cancel each other out, leading to the final simplified answer of 0.

Key concepts

Radicals in Mathematics

Understanding radicals is a fundamental aspect in algebra and higher mathematics. A radical, often represented by the square root symbol \( \sqrt{} \), signifies the operation of finding a number that, when multiplied by itself, gives the original number. For instance, if we consider \( \sqrt{9} \), we're looking for a number that, when squared, equals 9, which is 3.

When simplifying radicals that involve fractions, like \( \sqrt{\frac{9x}{16}} \), the process can be thought of as finding the square root of the numerator, \( \sqrt{9x} \), and the denominator, \( \sqrt{16} \), separately. Here, \( \sqrt{16} \) simplifies to 4, because \( 4^2 = 16 \). This approach of separating the numerator and the denominator under the radical helps in breaking down complex expressions into simpler forms, which can then be easily simplified or further manipulated if needed.

Fractional Exponents

Fractional exponents, also known as rational exponents, provide an alternative way of expressing radicals. The expression \( \sqrt{x} \) can also be written as \( x^{\frac{1}{2}} \). The denominator of the fraction indicates the root, in this case, 2 for a square root, and the numerator corresponds to the power to which the number is raised. This relationship can be extremely helpful in simplifying expressions involving square roots.

For instance, when starting with the expression \( \sqrt{\frac{9x}{16}} \) from the original exercise, converting to fractional exponents would yield \( (\frac{9x}{16})^{\frac{1}{2}} \). This highlights the connection between exponents and radicals, which is especially useful when dealing with more complex algebraic manipulations where applying the traditional radical form might be less intuitive or visually cumbersome.

Combining Like Terms

Combining like terms is an essential skill for simplifying algebraic expressions. Like terms are terms within an algebraic expression that have identical variable parts, such as coefficients or exponents. In our original expression, \( \frac{3\sqrt{x}}{4} \) and \( -\frac{3\sqrt{x}}{4} \) are considered like terms because they both contain the \( \sqrt{x} \) element.

When like terms are combined, their coefficients are added or subtracted from each other. If the coefficients are the same but have opposite signs, as in \( \frac{3}{4} \) and \( -\frac{3}{4} \) in this case, they cancel each other out, resulting in zero. This process simplifies the expression down to its most basic form and is an important step towards solving more complex equations.