Problem 25 - Algebra and Trigonometry

Question

Use the quotient rule to simplify the expressions Assume that \(x>0\). $$ \sqrt{\frac{49}{16}} $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{7}{4} \).

Step 1: Identify the Inner Expressions

The given expression can be seen as the square root of a fraction: \( \sqrt{\frac{49}{16}} \). The inner expressions that need to be squared are \(49\) in the numerator and \(16\) in the denominator.

Step 2: Apply the Square Root

The square root of \(49\) is \(7\) because \(7^2 = 49\). The square root of \(16\) is \(4\) because \(4^2 = 16\). This gives us the expression \( \frac{7}{4} \).

Step 3: Simplifying the Expression

The expression \( \frac{7}{4} \) is already in its simplest form, and there's no need to simplify it further.

Key Concepts

Quotient Rule for Square Roots

To effectively master algebra, understanding the quotient rule for square roots is essential. This rule allows us to simplify expressions involving the square root of a quotient, that is, a fraction. According to the quotient rule, for any nonnegative real numbers 'a' and 'b', provided that 'b' is not zero, the square root of the quotient \( \frac{a}{b} \) is equal to the quotient of the square roots of 'a' and 'b':
\[\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\].
It's important to note that this rule only applies when both 'a' and 'b' are greater than or equal to zero since the square root of a negative number is not a real number.

To apply this rule in practice, like in the exercise \( \sqrt{\frac{49}{16}} \), we separately take the square roots of the numerator and the denominator and then divide these square roots. By doing so, we simplify complex expressions into an understandable and easily manageable form.

Radical Expressions

Radical expressions, such as square roots, are omnipresent in algebra and need careful handling to solve them correctly. A radical expression is any expression that includes a radical symbol, with the most common being the square root symbol (\sqrt{}). To simplify radical expressions, you should look for factors that are perfect squares, which means they can be written as the square of some other integer.

For example, in the expression \( \sqrt{49} \), the number 49 is a perfect square because it is equal to \( 7^2 \). Recognizing this allows you to simplify \( \sqrt{49} \) to \( 7 \). Deeply understanding how to manipulate these expressions lets you effortlessly deal with more complex problems involving radicals, such as those with variables or higher-degree roots.

Square Root Simplification

Simplifying square roots is a process by which we break down a square root into its simplest possible form. If the number under the square root, known as the radicand, is a perfect square, then it can be expressed as the square of a whole number. If not, we seek to factor the radicand into a product of perfect squares and other numbers.

For the given exercise \( \sqrt{\frac{49}{16}} \), both 49 and 16 are perfect squares. Thus, we can simplify by taking the square root of each separately to get 7 and 4, respectively, followed by creating the quotient \( \frac{7}{4} \), which is in its simplest form. With practice, you become faster at identifying perfect squares and simplifying square roots, making it a crucial skill in higher-level mathematics.

Remember, the goal of square root simplification is to transform the root into a form that can be easily understood and manipulated for further mathematical processing or real-world application.