Problem 62 - Intermediate Algebra

Question

Use the quotient rule to divide. Then simplify if possible. See Example 5 $$ \frac{\sqrt{45}}{\sqrt{9}} $$

Step-by-Step Solution

Answer

\( \frac{\sqrt{45}}{\sqrt{9}} = \sqrt{5} \).

Step 1: Apply the Quotient Rule

The quotient rule for square roots states that \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \). Apply it to \( \frac{\sqrt{45}}{\sqrt{9}} \). This becomes \( \sqrt{\frac{45}{9}} \).

Step 2: Simplify the Division Inside the Square Root

Now perform the division inside the square root: \( \frac{45}{9} = 5 \). So now we have \( \sqrt{5} \).

Step 3: Simplify, If Possible

Check if \( \sqrt{5} \) can be simplified further. Since 5 is not a perfect square and has no square factors, \( \sqrt{5} \) is already in its simplest form.

Key Concepts

Simplifying Square Roots

Simplifying square roots involves reducing a square root expression to its simplest form. This typically means finding the prime factors of the number under the square root and pairing any duplicate factors. For example, when you simplify \( \sqrt{45} \), think about the factors of 45: 3, 3, and 5. Since there is a pair of 3s, we take one out of the square root, leaving \( 3\sqrt{5} \) because \( 3^2 = 9 \) and the number outside the root represents the paired factors.

However, when working directly with \( \sqrt{5} \), since 5 is a prime number, it has no paired factors. Therefore, \( \sqrt{5} \) is already simplified. Remember these two steps:

• Find the prime factors.
• Move any pairs of factors outside the square root.

It's crucial to ensure that the expression cannot be simplified further. In cases where no pairs exist or numbers are already prime, the square root is already in its simplest form.

Division of Radicals

Dividing radicals may seem daunting at first, but it becomes straightforward once you apply the quotient rule for square roots. The rule states that \( \frac{\sqrt{a}}{\sqrt{b}} \) can be rewritten as \( \sqrt{\frac{a}{b}} \). This means you can perform the division inside the square root before simplifying, making the process more manageable.

For instance, dividing \( \frac{\sqrt{45}}{\sqrt{9}} \) is simplified by first applying the quotient rule to get \( \sqrt{\frac{45}{9}} \), simplifying inside the root to \( \sqrt{5} \). This method efficiently reduces the complexity of handling separate radicals.

• Use the quotient rule to combine square roots.
• Perform division inside the square root.

This approach emphasizes simplification from inside the root first, ensuring the resultant expression is easily understandable and manageable.

Simplification of Radicals

Simplification of radicals is all about representing a radical expression in its most reduced form without altering its value. The process involves examining the expression under the radical sign, factoring it adequately, and ensuring any factors that can be squared are moved outside the radical sign.

For example, a complex radical like \( \sqrt{72} \) could initially seem cumbersome. However, decomposing 72 into its factors—\(2 \times 2 \times 2 \times 3 \times 3\) — allows us to simplify \( \sqrt{72} \) to \( 6\sqrt{2} \) by moving the pairs of twos and threes outside the radical.

• Break down the number into its prime factors.
• Remove pairs as single digits outside the radical.
• Leave non-paired factors inside the radical.

Always aim for a simple, concise expression that fully represents the number inside the radical, ensuring no further simplification is possible. This method not only simplifies calculation but also ensures clarity, avoiding unnecessary complexity in mathematical manipulation.