Problem 43 - Algebra 1: Concepts and Skills

Question

Simplify the expression. $$ \sqrt{\frac{10}{162}} $$

Step-by-Step Solution

Answer

The expression \(\sqrt{\frac{10}{162}}\) simplifies to \(\frac{\sqrt{5}}{9}\).

Step 1: Simplify the Fraction

To simplify the fraction \(\frac{10}{162}\), find the greatest common divisor (GCD) of 10 and 162. In this case, the GCD is 2. Divide both the numerator and the denominator by 2: \(\frac{10}{2} = 5\) and \(\frac{162}{2}=81\), so we rewrite the expression as \(\sqrt{\frac{5}{81}}\).

Step 2: Simplify the Square Root

We can simplify the square root by taking the square root of the numerator and the denominator separately. The square root of 5 cannot be simplified further, but the square root of 81 is 9. So we can rewrite the expression as \(\frac{\sqrt{5}}{9}\).

Step 3: Final Answer

So, after simplifying, the given expression \(\sqrt{\frac{10}{162}}\) simplifies to \(\frac{\sqrt{5}}{9}\).

Key Concepts

Greatest Common Divisor (GCD)

When you're simplifying fractions, one of the first steps often involves finding the greatest common divisor (GCD). The GCD between two numbers is the largest positive integer that evenly divides both numbers.

Here's how you find it:

• List all the divisors of both numbers.
• Identify the common factors.
• The largest of these common factors is the GCD.

In the original exercise, we looked at the numbers 10 and 162.

• The divisors of 10 are 1, 2, 5, and 10.
• The divisors of 162 are 1, 2, 3, 6, 9, 18, 27, 54, 81, and 162.

The number 2 is the largest number common to both lists. Therefore, our GCD is 2. By dividing both the numerator and denominator by this GCD, we make the fraction simpler and more manageable.

Simplifying Fractions

Once you have the GCD, it becomes straightforward to simplify fractions. It involves dividing both the numerator and the denominator by their greatest common divisor.

Simplifying fractions helps in reducing calculations and gives a clearer understanding of their relationship.

• Take the GCD of the fraction’s numerator and denominator.
• Divide both the top and the bottom of the fraction by this number.
• Write the new fraction.

In the step-by-step solution, we divided the fraction \( \frac{10}{162} \) by its GCD, 2. This led us to \( \frac{5}{81} \).

This form is simplified, but sometimes, like in this exercise, you will need to continue simplifying using other mathematical operations, like taking square roots.

Square Roots

Square roots allow us to determine a number which, when multiplied by itself, gives the original number. A square root can be simplified by identifying perfect squares within it.

The expression \( \sqrt{\frac{5}{81}} \) simplifies differently.

• The numerator, 5, is a prime number and doesn't have a perfect square factor other than 1.
• For the denominator, \(81\), it is already a perfect square because \(9 \times 9 = 81\).

Thus:

• Leave \( \sqrt{5} \) as is because it does not simplify.
• Change \( \sqrt{81} \) to 9, since it's a perfect square.

The result is a simplified fractional square root \( \frac{\sqrt{5}}{9} \). It's important to simplify whenever possible for cleaner results.