Problem 25 - Algebra 2

Question

Simplify. $$ \sqrt{\frac{350}{81}} $$

Step-by-Step Solution

Answer

\(\frac{5\sqrt{14}}{9}\)

Step 1: Break Down the Square Root

Start by applying the property of square roots to the expression: \(\sqrt{\frac{350}{81}} = \frac{\sqrt{350}}{\sqrt{81}}\).

Step 2: Simplify the Denominator

The square root of 81 is 9, since \(9^2 = 81\). Thus, \(\sqrt{81} = 9\). So, the expression becomes \(\frac{\sqrt{350}}{9}\).

Step 3: Simplify the Numerator

The square root of 350 can be simplified by finding its prime factors. Factor 350 into 2, 5, and 7: \(350 = 2 \times 5^2 \times 7\). The square root is \(\sqrt{350} = \sqrt{2 \times 5^2 \times 7} = 5\sqrt{14}\).

Step 4: Combine the Simplified Parts

Combine the results to get \(\frac{5\sqrt{14}}{9}\). This is the simplified form of the original expression.

Key Concepts

Prime Factorization

To simplify expressions involving square roots, especially when dealing with integers, prime factorization is an essential technique. Prime factorization is the process of breaking down a number into its basic building blocks, which are prime numbers. These are numbers greater than 1 that have no divisors other than 1 and themselves.

For example, let's look at the number 350 as mentioned in our exercise. The prime factorization strategy involves continuously dividing the number by the smallest prime until only prime numbers remain.

• Start with the smallest prime, 2. Since 350 is even, divide by 2: 350 ÷ 2 = 175.
• Next, use 5, the next smallest prime: 175 ÷ 5 = 35.
• Continue with 5 again: 35 ÷ 5 = 7.
• Finally, 7 is already a prime number.

Thus, 350 can be expressed as a product of prime numbers: 2 x 5^2 x 7. Recognizing repeated factors is crucial for simplifying square roots later on.

Square Root Properties

Square root properties provide powerful shortcuts that make simplifying expressions much easier. One important property is that the square root of a fraction can be separated into the square root of its numerator and the square root of its denominator. Mathematically, this is written as: \[\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \]

This property allows us to handle the numerator and denominator separately, breaking what might seem like a complex expression into more manageable parts.

Another useful property relates to the square root of a product. If you have two factors within a square root, they can be evaluated separately: \[\sqrt{a \times b} = \sqrt{a} \times \sqrt{b} \]

Using this, square roots of perfect squares become easy to manage because the square root of a perfect square is simply the number that was squared. So, using this property, simplifying \(\sqrt{81}\) gives us 9, because 9 x 9 = 81! These properties help us navigate through square roots efficiently.

Fraction Simplification

Fraction simplification is fundamental in mathematics because it helps in reducing fractions to their simplest form. A fraction consists of a numerator and a denominator. To simplify it means to express it in its most reduced terms where both parts share no common factors other than 1.

Let's simplify \(\frac{5\sqrt{14}}{9}\) as our final expression. Here, barring any common factors or need for further evaluation of \(\sqrt{14}\), the fraction is already in its simplest form when compared to the original more complex expression. There's no common factor between 5 and 9 apart from 1, and \(\sqrt{14}\) doesn't simplify further since 14 cannot be simplified to a perfect square.

Always check for any common factors. If a numerator and denominator share a common factor (besides 1), divide both by that factor to reach the simplest form. Remember, being attentive about such factors ensures an expression is as reduced as possible.