Problem 53 - Algebra 2

Question

Simplify. Divide \(\sqrt{14}\) by \(\sqrt{35}\)

Step-by-Step Solution

Answer

\(\sqrt{\frac{2}{5}}\)

Step 1: Write the Expression Using Fractions

First, we need to express the division operation of two square roots as a fraction. So, divide the square root of 14 by the square root of 35: \[\frac{\sqrt{14}}{\sqrt{35}}.\]

Step 2: Use the Quotient Property of Square Roots

The quotient property of square roots states that \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \). So, apply this property to our expression:\[\sqrt{\frac{14}{35}}.\]

Step 3: Simplify the Fraction Inside the Square Root

Now, simplify the fraction under the square root by finding the greatest common divisor (GCD) of 14 and 35. The GCD is 7, so:\[\frac{14}{35} = \frac{14 \div 7}{35 \div 7} = \frac{2}{5}.\]

Step 4: Express the Result in Simplified Form

Substitute the simplified fraction back into the square root:\[\sqrt{\frac{2}{5}}.\]This is the simplified form of the original expression.

Key Concepts

Quotient Property of Square Roots

When dealing with square roots, the quotient property is a useful rule that can simplify many expressions. It tells us that when we divide one square root by another, we can write it as a single square root of the quotient of the numbers inside. Mathematically, this is expressed as \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \). This property works because square roots are a type of exponent. The property of exponents tells us that when we divide similar bases, we subtract the exponents. Since square roots are equivalent to raising a number to the power of one-half, the property allows these operations. Using this property can make calculations easier and more straightforward, turning a complex division into a simple square root of a fraction. Let's see how it applies in practice!

Greatest Common Divisor (GCD)

The concept of the greatest common divisor (GCD) is crucial for simplifying fractions, especially under radicals. The GCD of two numbers is the largest number that divides both of them without leaving a remainder. To determine the GCD:

• List the factors of each number.
• Identify the greatest factor that appears in both lists.

For example, to simplify \( \frac{14}{35} \), we find the GCD of 14 and 35. The factors of 14 are 1, 2, 7, and 14. For 35, the factors are 1, 5, 7, and 35. Here, 7 is the greatest common factor. By dividing both the numerator and the denominator by their GCD, we simplify the fraction to \( \frac{2}{5} \). Using the GCD not only helps in simplifying fractions but also ensures the result is in its simplest form, which makes further calculations cleaner and more manageable.

Simplification of Fractions

Simplifying fractions is an important skill in mathematics that allows us to express the fraction in its simplest form. This means using the smallest possible numerator and denominator while maintaining the value of the fraction. To simplify a fraction:

• Determine the GCD of the numerator and the denominator.
• Divide both the top and bottom of the fraction by this number.

Take \( \frac{14}{35} \) as an example. We've already calculated the GCD as 7. By dividing both the numerator and the denominator by 7, we get \( \frac{2}{5} \). This is the simplified form. Simplifying is especially useful when dealing with expressions under the square root, as it makes working out the square root more straightforward. When fractions are simplified, they are easier to understand, calculate, and compare with other numbers or fractions. Mastering this process is key to advancing in more complex mathematics.