Problem 63 - Algebra for College Students

Question

Change each radical to simplest radical form. \(\frac{-8 \sqrt{18}}{10 \sqrt{50}}\)

Step-by-Step Solution

Answer

The simplest form is \( \frac{-12}{25} \).

Step 1: Simplify the Radicals

Start by simplifying each radical. The radical \( \sqrt{18} \) can be expressed as \( \sqrt{9 \times 2} = \sqrt{9} \cdot \sqrt{2} = 3 \sqrt{2} \). Similarly, the radical \( \sqrt{50} \) can be expressed as \( \sqrt{25 \times 2} = \sqrt{25} \cdot \sqrt{2} = 5 \sqrt{2} \).

Step 2: Substitute the Simplified Radicals

Substitute the simplified values back into the original expression: \(\frac{-8 \sqrt{18}}{10 \sqrt{50}} = \frac{-8 \cdot 3 \sqrt{2}}{10 \cdot 5 \sqrt{2}}.\)

Step 3: Simplify the Fraction

Now, simplify the fractional expression: Multiply to get inside: \(= \frac{-24 \sqrt{2}}{50 \sqrt{2}}. \)Since \(\sqrt{2}\) is present in both the numerator and the denominator, you can cancel it out: \(= \frac{-24}{50}. \)

Step 4: Reduce the Fraction

The fraction \( \frac{-24}{50} \) needs to be reduced to its simplest form. Find the greatest common divisor (GCD) of 24 and 50, which is 2:\(= \frac{-24 \div 2}{50 \div 2} = \frac{-12}{25}.\)

Step 5: Final Result

Now we have the simplest radical form of the given expression: \(\frac{-12}{25}. \) Thus, the simplest radical form of the expression \( \frac{-8 \sqrt{18}}{10 \sqrt{50}} \) is \( \frac{-12}{25}. \)

Key Concepts

Simplifying Radicals

When simplifying radicals, we aim to express a radical in its simplest form. This involves breaking down the number under the square root (the radicand) into its prime factors. Next, identify perfect squares within these factors and make use of the property that the square root of a perfect square simplifies to a whole number.

For instance, consider \( \sqrt{18} \). The number 18 can be factored as \( 9 \times 2 \), where 9 is a perfect square. Therefore, \( \sqrt{18} \) can be simplified to \( \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2} \). Simplifying radicals makes them more manageable and sets the stage for further operations like multiplication or division.

Factoring Under Radicals

Factoring under radicals involves breaking down the number inside the radical sign into its simplest constituents. This step is crucial for simplification as it helps reveal components that can be extracted from the radical.

To factor a number under a radical, identify its prime factors. For example, \( 50 \) can be expressed as \( 25 \times 2 \). Since 25 is a perfect square, \( \sqrt{50} \) becomes \( \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2} \). Factoring correctly will lead you to simpler expressions which are easier to work with in calculations.

Cancelling Common Factors

Cancelling common factors is a straightforward but powerful technique in simplifying expressions, especially fractions involving radicals. When you have the same factor in both the numerator and denominator, you can cancel it out, leaving a simpler expression.

In the expression \( \frac{-24 \sqrt{2}}{50 \sqrt{2}} \), the \( \sqrt{2} \) present in both the numerator and the denominator can be cancelled. This results in \( \frac{-24}{50} \), significantly simplifying the process by removing the radical altogether.

Fraction Reduction

Fraction reduction refers to simplifying a fraction to its lowest terms. This process requires dividing both the numerator and the denominator by their greatest common factor until no further reduction is possible.

For example, the fraction \( \frac{-24}{50} \) can be reduced by identifying common factors. Start by recognizing that both numbers are divisible by 2. Dividing both the numerator and denominator by 2, we get \( \frac{-12}{25} \). Ensuring the fraction is in its lowest terms makes it easier to understand and compare with other values.

Greatest Common Divisor (GCD)

The greatest common divisor (GCD) is a key concept that allows us to simplify fractions efficiently. The GCD of two numbers is the largest number that divides both of them without any remainder. Finding the GCD helps in reducing fractions to their simplest form.

To find the GCD of 24 and 50, list the divisors of each number:

• Divisors of 24: 1, 2, 3, 4, 6, 8, 12, 24
• Divisors of 50: 1, 2, 5, 10, 25, 50

The largest common divisor is 2. Using the GCD to divide both parts of the fraction \( \frac{-24}{50} \), achieves the simplest form \( \frac{-12}{25} \). Understanding and applying the GCD ensures that solutions are presented in the most precise and concise form.