Problem 25 - College Algebra Essentials

Question

Use the quotient rule to simplify the expressions in Exercises \(23-32\) Assume that \(x>0\) $$\sqrt{\frac{49}{16}}$$

Step-by-Step Solution

Answer

The simplified form of the expression \(\sqrt{\frac{49}{16}}\) is \(\frac{7}{4}\).

Step 1: Usages of Square Root Property

The square root of a quotient is equal to the quotient of the square roots. Therefore, we apply this property to the expression. So, we express \(\sqrt{\frac{49}{16}}\) as \(\frac{\sqrt{49}}{\sqrt{16}}\).

Step 2: Calculating the Square Root

Calculate the square root of the numerator and denominator separately. The square root of 49 is 7 and the square root of 16 is 4.

Step 3: Final Simplification

Substituting these values into the expression gives \(\frac{7}{4}\) as the final simplified form.

Key Concepts

Square Root Property

When dealing with square roots of fractions, understanding the square root property is essential. The square root property allows us to simplify expressions where a square root is applied to a fraction by expressing the square root of a quotient as the quotient of the square roots. This means that for any positive values of \(a\) and \(b\), the square root of a fraction \(\frac{a}{b}\) can be expressed as \(\frac{\sqrt{a}}{\sqrt{b}}\). This property significantly simplifies the process as it lets us work on the numerator and the denominator separately, which makes calculations easier.

Simplifying Expressions

Simplifying expressions is about making them easier to understand or execute without changing their value. When simplifying expressions like \(\sqrt{\frac{49}{16}}\), we can break down the process into manageable steps using mathematical properties. By using the square root property, we first separate the expression into two simpler square roots: \(\sqrt{49}\) and \(\sqrt{16}\). Simplifying each of these individually allows us to find the simplest form, which is easy to analyze and often required for further calculations or evaluations. Keeping expressions in their simplest form is beneficial for clarity and precision in math.

Square Root Calculation

Calculating square roots is a foundational math skill involving finding a number that, when multiplied by itself, gives the original number. For perfect squares like 49 and 16, this is straightforward.

• For the numerator 49, the square root is 7, since \(7 \times 7 = 49\).
• For the denominator 16, the square root is 4, because \(4 \times 4 = 16\).

Using these results, we substitute them back into the expression \(\frac{\sqrt{49}}{\sqrt{16}}\) to get \(\frac{7}{4}\). This simplification makes it easier to understand and to use in further mathematical operations or practical applications.