How Do You Simplify Square Root Expressions: A Comprehensive Guide

Understanding how to simplify square root expressions is crucial in various mathematical applications. The goal is to make the number inside the square root as small as possible while ensuring it remains a whole number.

Product Property of Square Roots

The product property states that the square root of a product is equal to the product of the square roots:

\(\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}\)

Quotient Property of Square Roots

The quotient property states that the square root of a quotient is equal to the quotient of the square roots:

\(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\)

Examples of Simplifying Square Roots

Example 1: Simplify \(\sqrt{12}\)

1. Identify factors of 12 that include a perfect square: \(12 = 4 \times 3\)
2. Apply the product property: \(\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \cdot \sqrt{3}\)
3. Simplify: \(\sqrt{4} = 2\), so \(\sqrt{12} = 2\sqrt{3}\)

Example 2: Simplify \(\sqrt{45}\)

1. Identify factors of 45 that include a perfect square: \(45 = 9 \times 5\)
2. Apply the product property: \(\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \cdot \sqrt{5}\)
3. Simplify: \(\sqrt{9} = 3\), so \(\sqrt{45} = 3\sqrt{5}\)

Example 3: Simplify \(\sqrt{72}\)

1. Identify factors of 72 that include a perfect square: \(72 = 36 \times 2\)
2. Apply the product property: \(\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \cdot \sqrt{2}\)
3. Simplify: \(\sqrt{36} = 6\), so \(\sqrt{72} = 6\sqrt{2}\)

Simplifying Square Roots Involving Fractions

Example 4: Simplify \(\sqrt{\frac{36}{4}}\)

1. Apply the quotient property: \(\sqrt{\frac{36}{4}} = \frac{\sqrt{36}}{\sqrt{4}}\)
2. Simplify: \(\sqrt{36} = 6\) and \(\sqrt{4} = 2\), so \(\sqrt{\frac{36}{4}} = \frac{6}{2} = 3\)

General Steps to Simplify Square Root Expressions

• Factor the number inside the square root into its prime factors.
• Identify and separate perfect squares from the factors.
• Apply the product property to separate the square roots of the perfect squares and the remaining factors.
• Simplify the square roots of the perfect squares.

Sample Problems

Simplifying square roots involves breaking down the radicand (the number inside the square root) into its prime factors to find perfect squares. This process makes the expression easier to understand and work with. Understanding the properties of square roots and their rules is essential in simplifying these expressions effectively.

When simplifying square roots, we apply the following steps:

1. Factor the radicand: Break down the number inside the square root into its prime factors. For example, for √72, the prime factors are 2, 2, 2, 3, and 3.
2. Identify perfect squares: Look for pairs of prime factors that form perfect squares. In the example of √72, the pairs are 2×2 and 3×3, which are perfect squares.
3. Rewrite the expression: Express the radicand as the product of its perfect squares and any remaining factors. For √72, this becomes √(2×2×3×3×2).
4. Simplify the square root: Take the square root of the perfect squares and multiply them together, leaving any remaining factors under the square root. For √72, this simplifies to 6√2.

By following these steps, we can simplify square roots systematically, making them easier to work with in various mathematical contexts.

Understanding how to simplify square root expressions is foundational in algebra and higher mathematics. Simplifying square roots involves a few basic concepts and rules, which we will outline step-by-step.

Product Property of Square Roots

One of the key properties is the product property, which states:

\[
\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}
\]

This property allows you to break down a complex square root into simpler components.

Quotient Property of Square Roots

Another important property is the quotient property, which is expressed as:

\[
\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}
\]

This property is particularly useful when dealing with fractions under the square root sign.

Steps to Simplify Square Roots

1. Simplify the fraction inside the radicand (the expression under the square root), if possible.
2. Use the quotient property to separate the square root of the numerator and the square root of the denominator.
3. Break down each square root into its prime factors.
4. Pair up the prime factors and move each pair outside the square root sign.

Examples

Common Mistakes to Avoid

• Do not separate sums inside a square root. For example, \(\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}\).
• Always simplify the fraction inside the radicand before applying the quotient property.
• Ensure all factors are in their simplest form before moving pairs outside the square root sign.

Simplifying square roots without fractions involves breaking down the radicand (the number under the square root) into its prime factors and simplifying using the properties of square roots. Here are the steps to simplify square roots without fractions:

1. Factor the radicand into its prime factors: Break down the number under the square root into its prime factors. For example, the prime factors of 75 are 5 and 3 (since 75 = 5^2 * 3).

2. Group the factors into pairs: Identify and group the pairs of prime factors. Each pair of identical factors can be taken out of the square root as a single factor. For example, for √75, you have the pair (5^2) and 3, so √75 = √(5^2 * 3).

3. Take out the pairs from under the square root: For each pair of identical factors, take one factor out of the square root. In our example, √(5^2 * 3) becomes 5√3.

4. Simplify the expression: Multiply the factors taken out of the square root by any remaining factors still under the square root. In the example, you get 5√3 as the simplified form of √75.

Let's look at some examples to illustrate these steps:

Examples

• Example 1: Simplify √72.

  Step 1: Factor the radicand into its prime factors: 72 = 2^3 * 3^2.

  Step 2: Group the factors into pairs: √(2^3 * 3^2) = √(2^2 * 2 * 3^2).

  Step 3: Take out the pairs from under the square root: 2 * 3√2.

  Step 4: Simplify the expression: 6√2.

• Example 2: Simplify √50.

  Step 1: Factor the radicand into its prime factors: 50 = 2 * 5^2.

  Step 2: Group the factors into pairs: √(2 * 5^2).

  Step 3: Take out the pairs from under the square root: 5√2.

  Step 4: Simplify the expression: 5√2.

By following these steps, you can simplify any square root expression that does not involve fractions.

Simplifying square roots involves expressing the square root in its simplest form by factoring out perfect squares. Here are some step-by-step examples to illustrate the process:

Example 1: Simplify √12

1. Identify the factors of 12 that include a perfect square: 12 = 4 × 3.
2. Apply the square root to each factor: √12 = √(4 × 3).
3. Simplify the square root of the perfect square: √4 = 2.
4. Combine the simplified term with the remaining square root: √12 = 2√3.

Example 2: Simplify √45

1. Identify the factors of 45 that include a perfect square: 45 = 9 × 5.
2. Apply the square root to each factor: √45 = √(9 × 5).
3. Simplify the square root of the perfect square: √9 = 3.
4. Combine the simplified term with the remaining square root: √45 = 3√5.

Example 3: Simplify √18

1. Identify the factors of 18 that include a perfect square: 18 = 9 × 2.
2. Apply the square root to each factor: √18 = √(9 × 2).
3. Simplify the square root of the perfect square: √9 = 3.
4. Combine the simplified term with the remaining square root: √18 = 3√2.

Example 4: Simplify √50

1. Identify the factors of 50 that include a perfect square: 50 = 25 × 2.
2. Apply the square root to each factor: √50 = √(25 × 2).
3. Simplify the square root of the perfect square: √25 = 5.
4. Combine the simplified term with the remaining square root: √50 = 5√2.

Example 5: Simplify √72

1. Identify the factors of 72 that include a perfect square: 72 = 36 × 2.
2. Apply the square root to each factor: √72 = √(36 × 2).
3. Simplify the square root of the perfect square: √36 = 6.
4. Combine the simplified term with the remaining square root: √72 = 6√2.

By following these steps, you can simplify square roots without fractions, ensuring the expression is in its simplest form.

Simplifying square roots that involve fractions can be made easier by using the Quotient Property of Square Roots. This property states that the square root of a quotient is equal to the quotient of the square roots of the numerator and the denominator:

\(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\)

Here are the steps to simplify square roots with fractions:

1. Simplify the fraction inside the radical (if possible): If the fraction inside the square root can be reduced, do so first. This will make the subsequent steps easier.

2. Apply the Quotient Property: Rewrite the square root of the fraction as the quotient of the square roots of the numerator and the denominator.

   \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\)

3. Simplify the square roots: If possible, simplify the square roots in the numerator and the denominator.

Let's look at a few examples:

Example 1:

Simplify: \(\sqrt{\frac{25}{49}}\)

1. Simplify the fraction: \(\frac{25}{49}\) is already in its simplest form.
2. Apply the Quotient Property: \(\sqrt{\frac{25}{49}} = \frac{\sqrt{25}}{\sqrt{49}}\)
3. Simplify the square roots: \(\frac{\sqrt{25}}{\sqrt{49}} = \frac{5}{7}\)

So, \(\sqrt{\frac{25}{49}} = \frac{5}{7}\).

Example 2:

Simplify: \(\sqrt{\frac{18}{50}}\)

1. Simplify the fraction: \(\frac{18}{50} = \frac{9}{25}\)
2. Apply the Quotient Property: \(\sqrt{\frac{9}{25}} = \frac{\sqrt{9}}{\sqrt{25}}\)
3. Simplify the square roots: \(\frac{\sqrt{9}}{\sqrt{25}} = \frac{3}{5}\)

So, \(\sqrt{\frac{18}{50}} = \frac{3}{5}\).

Example 3:

Simplify: \(\sqrt{\frac{50x^2}{18y^4}}\)

1. Simplify the fraction: \(\frac{50x^2}{18y^4} = \frac{25x^2}{9y^4}\)
2. Apply the Quotient Property: \(\sqrt{\frac{25x^2}{9y^4}} = \frac{\sqrt{25x^2}}{\sqrt{9y^4}}\)
3. Simplify the square roots: \(\frac{\sqrt{25x^2}}{\sqrt{9y^4}} = \frac{5x}{3y^2}\)

So, \(\sqrt{\frac{50x^2}{18y^4}} = \frac{5x}{3y^2}\).

By following these steps and applying the Quotient Property, you can simplify square roots involving fractions effectively.

The quotient property of square roots states that the square root of a quotient is equal to the quotient of the square roots of the numerator and the denominator. In mathematical terms, if \(a\) and \(b\) are non-negative real numbers and \(b \ne 0\), then:

\[\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\]

This property is useful when simplifying square roots of fractions. Here’s a step-by-step guide to using the quotient property:

1. Simplify the fraction inside the radical, if possible.

2. Use the quotient property to rewrite the radical as the quotient of two separate radicals.

3. Simplify the radicals in the numerator and the denominator separately.

Let's look at some examples to understand this better:

Example 1: Simplify \(\sqrt{\frac{25}{9}}\)

\[\sqrt{\frac{25}{9}} = \frac{\sqrt{25}}{\sqrt{9}} = \frac{5}{3}\]

Both 25 and 9 are perfect squares, so their square roots are straightforward.

Example 2: Simplify \(\sqrt{\frac{50}{4}}\)

First, simplify the fraction inside the radical:
\[\sqrt{\frac{50}{4}} = \sqrt{12.5}\]
Next, use the quotient property:
\[\sqrt{12.5} = \sqrt{\frac{25}{2}} = \frac{\sqrt{25}}{\sqrt{2}} = \frac{5}{\sqrt{2}}\]
Finally, rationalize the denominator:
\[\frac{5}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{5\sqrt{2}}{2}\]

Example 3: Simplify \(\sqrt{\frac{72x^4}{8x^2}}\)

Simplify the fraction inside the radical:
\[\sqrt{\frac{72x^4}{8x^2}} = \sqrt{9x^2} = 3x\]
In this case, the fraction simplifies completely before applying the quotient property.

By following these steps, you can simplify square roots of fractions effectively. The quotient property is a powerful tool that helps in breaking down complex square root expressions into simpler forms.

Simplifying square roots with fractions involves applying the quotient property of square roots. Here are several detailed examples to illustrate the process:

1. Simplify: \(\sqrt{\frac{21}{64}}\)

   1. Apply the quotient property: \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\)
   2. Rewrite the expression: \(\frac{\sqrt{21}}{\sqrt{64}}\)
   3. Simplify the square root of 64: \(\frac{\sqrt{21}}{8}\)

2. Simplify: \(\sqrt{\frac{45x^5}{y^4}}\)

   1. Apply the quotient property: \(\frac{\sqrt{45x^5}}{\sqrt{y^4}}\)
   2. Simplify the square roots separately: \(\frac{\sqrt{9x^4} \cdot \sqrt{5x}}{\sqrt{y^4}}\)
   3. Rewrite as: \(\frac{3x^2 \sqrt{5x}}{y^2}\)

3. Simplify: \(\sqrt{\frac{54u^7}{v^8}}\)

   1. Apply the quotient property: \(\frac{\sqrt{54u^7}}{\sqrt{v^8}}\)
   2. Simplify the square roots separately: \(\frac{\sqrt{9u^6} \cdot \sqrt{6u}}{\sqrt{v^8}}\)
   3. Rewrite as: \(\frac{3u^3 \sqrt{6u}}{v^4}\)

4. Simplify: \(\sqrt{\frac{80m^3}{n^6}}\)

   1. Apply the quotient property: \(\frac{\sqrt{80m^3}}{\sqrt{n^6}}\)
   2. Simplify the square roots separately: \(\frac{\sqrt{16m^2} \cdot \sqrt{5m}}{\sqrt{n^6}}\)
   3. Rewrite as: \(\frac{4m \sqrt{5m}}{n^3}\)

5. Simplify: \(\sqrt{\frac{27m^3}{196}}\)

   1. Simplify the fraction inside the radical: \(\frac{27m^3}{196}\) cannot be simplified.
   2. Apply the quotient property: \(\frac{\sqrt{27m^3}}{\sqrt{196}}\)
   3. Simplify the square roots separately: \(\frac{\sqrt{9m^2} \cdot \sqrt{3m}}{\sqrt{196}}\)
   4. Rewrite as: \(\frac{3m \sqrt{3m}}{14}\)

These examples demonstrate the systematic approach to simplifying square roots involving fractions by using the quotient property. Each step involves breaking down the expression into more manageable parts and simplifying where possible.

When simplifying square roots, there are several common mistakes that can lead to incorrect results. Here are some key pitfalls to avoid and tips for getting it right:

• Forgetting to Check for Perfect Squares:

  Always check if the number under the square root is a perfect square. This is the simplest form of simplification. For example, \( \sqrt{16} \) simplifies directly to 4.

• Incorrectly Applying the Product Rule:

  When using the property \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \), ensure that both \( a \) and \( b \) are non-negative. Misapplying this rule can lead to incorrect simplifications.

• Not Simplifying the Fraction First:

  When dealing with square roots of fractions, simplify the fraction first if possible. For example, \( \sqrt{\frac{4}{9}} = \frac{\sqrt{4}}{\sqrt{9}} = \frac{2}{3} \).

• Ignoring Prime Factorization:

  Breaking down the number into its prime factors can make simplification easier. For instance, \( \sqrt{72} \) can be simplified by recognizing that 72 = 2 × 2 × 2 × 3 × 3, which simplifies to \( 6\sqrt{2} \).

• Rushing Through the Process:

  Simplifying square roots requires careful step-by-step work. Rushing through can lead to errors. Take your time and double-check each step.

Additional Tips:

• Use a calculator to check your work if you're unsure about a simplification.
• Practice regularly with different types of square root problems to build confidence and proficiency.

Example Problems:

1. Simplify \( \sqrt{50} \):

\[
\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}
\]

2. Simplify \( \sqrt{\frac{18}{2}} \):

\[
\sqrt{\frac{18}{2}} = \sqrt{9} = 3
\]

By being aware of these common mistakes and following these guidelines, you can improve your accuracy and efficiency in simplifying square root expressions.

When simplifying square root expressions, advanced techniques can help you handle more complex scenarios. Here are some methods to consider:

Simplifying Nested Radicals

Nested radicals involve square roots within square roots. To simplify them, you need to find a way to express the nested radical in a simpler form.

For example, consider the expression:

\[\sqrt{2 + \sqrt{3}}\]

To simplify, assume it can be written as \(\sqrt{a} + \sqrt{b}\). Squaring both sides and equating terms will help determine \(a\) and \(b\).

Using Conjugates

Rationalizing the denominator often involves using conjugates. This technique is particularly useful when dealing with binomials in the denominator.

For example:

\[\frac{1}{\sqrt{5} + \sqrt{3}}\]

Multiply the numerator and denominator by the conjugate of the denominator:

\[\frac{1}{\sqrt{5} + \sqrt{3}} \times \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} - \sqrt{3}} = \frac{\sqrt{5} - \sqrt{3}}{(\sqrt{5})^2 - (\sqrt{3})^2} = \frac{\sqrt{5} - \sqrt{3}}{5 - 3} = \frac{\sqrt{5} - \sqrt{3}}{2}\]

Simplifying Expressions with Variables

When simplifying square roots of expressions with variables, assume the variables represent non-negative numbers.

For example, simplify \(\sqrt{50x^4y^6}\):

Step 1: Factor inside the radical:

\[\sqrt{50x^4y^6} = \sqrt{25 \cdot 2 \cdot (x^2)^2 \cdot (y^3)^2}\]

Step 2: Take the square root of each factor:

\[\sqrt{25} \cdot \sqrt{2} \cdot \sqrt{(x^2)^2} \cdot \sqrt{(y^3)^2} = 5 \cdot \sqrt{2} \cdot x^2 \cdot y^3 = 5x^2y^3\sqrt{2}\]

Handling Complex Numbers

Square roots of negative numbers involve complex numbers. The imaginary unit \(i\) is used where \(i^2 = -1\).

For example, simplify \(\sqrt{-9}\):

Since \(\sqrt{-1} = i\) and \(\sqrt{9} = 3\), we have:

\[\sqrt{-9} = \sqrt{9} \cdot \sqrt{-1} = 3i\]

Combining Radicals

To combine radicals, they must have the same radicand. Use properties of exponents to simplify and combine.

For example:

\[\sqrt{8} + \sqrt{18} = \sqrt{4 \cdot 2} + \sqrt{9 \cdot 2} = 2\sqrt{2} + 3\sqrt{2} = 5\sqrt{2}\]

By applying these advanced techniques, you can simplify a wide range of square root expressions efficiently and accurately.

In this section, we will explore advanced techniques for simplifying square root expressions. These techniques often involve more complex algebraic manipulations and the use of multiple properties of square roots. Here are some detailed examples:

Example 1: Simplifying Nested Square Roots

Consider the expression \( \sqrt{20 \times \sqrt{5 \times \sqrt{2}}} \).

1. First, simplify the innermost square root: \( \sqrt{2} \).
2. Multiply it by 5: \( 5 \times \sqrt{2} = \sqrt{25 \times 2} = 5\sqrt{2} \).
3. Now, simplify \( \sqrt{5\sqrt{2}} \) within the original expression: \( \sqrt{20 \times 5\sqrt{2}} = \sqrt{100 \times 2} = 10\sqrt{2} \).

Thus, \( \sqrt{20 \times \sqrt{5 \times \sqrt{2}}} = 10\sqrt{2} \).

Example 2: Simplifying Expressions with Variables

Consider the expression \( \sqrt{50x^4y^6} \).

1. Factor the numerical part: \( 50 = 2 \times 25 \), so \( \sqrt{50} = \sqrt{2 \times 25} = 5\sqrt{2} \).
2. Apply the square root to the variables separately:
   • \( x^4 \) becomes \( x^2 \) outside the square root.
   • \( y^6 \) becomes \( y^3 \) outside the square root.
3. Combine these results: \( \sqrt{50x^4y^6} = 5x^2y^3\sqrt{2} \).

Example 3: Simplifying Complex Fractions

Consider the expression \( \frac{\sqrt{75}}{\sqrt{3}} \).

1. Combine the square roots: \( \frac{\sqrt{75}}{\sqrt{3}} = \sqrt{\frac{75}{3}} \).
2. Simplify the fraction inside the square root: \( \frac{75}{3} = 25 \).
3. Apply the square root: \( \sqrt{25} = 5 \).

Thus, \( \frac{\sqrt{75}}{\sqrt{3}} = 5 \).

Example 4: Using the Product Rule

Consider the expression \( \sqrt{18 \times 8} \).

1. First, factor the numbers inside the square root: \( 18 = 2 \times 9 \) and \( 8 = 2^3 \).
2. Combine and simplify: \( \sqrt{18 \times 8} = \sqrt{(2 \times 9) \times (2^3)} = \sqrt{2^4 \times 9} = \sqrt{16 \times 9} \).
3. Apply the square roots: \( \sqrt{16} = 4 \) and \( \sqrt{9} = 3 \).
4. Multiply the results: \( 4 \times 3 = 12 \).

Thus, \( \sqrt{18 \times 8} = 12 \).

Example 5: Simplifying Expressions with Mixed Terms

Consider the expression \( 3\sqrt{48} + 2\sqrt{75} \).

1. Simplify each term separately:
   • \( \sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3} \).
   • \( \sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3} \).
2. Substitute these back into the expression: \( 3(4\sqrt{3}) + 2(5\sqrt{3}) \).
3. Combine like terms: \( 12\sqrt{3} + 10\sqrt{3} = 22\sqrt{3} \).

Thus, \( 3\sqrt{48} + 2\sqrt{75} = 22\sqrt{3} \).

Simplifying square root expressions has numerous practical applications in various fields. Here are some examples:

1. Physics and Engineering

In physics and engineering, square roots are often used in formulas involving acceleration, velocity, and gravitational force. For example:

• The time \( t \) it takes for an object to fall from a height \( h \) is given by \( t = \frac{\sqrt{h}}{4} \). If an object is dropped from a height of 64 feet, the time to hit the ground is calculated as:
• \[\sqrt{64} = 8\]
• \[\frac{8}{4} = 2 \text{ seconds}\]

2. Construction and Architecture

Square roots are used to determine dimensions in construction projects. For example, to find the side length of a square area:

• If a square lawn covers 370 square feet, the side length \( s \) is given by:
• \[s = \sqrt{370} \approx 19.2 \text{ feet}\]

3. Accident Investigation

Accident investigators use square roots to estimate the speed of vehicles before they applied brakes based on skid marks. The speed \( v \) in mph can be found using the length \( d \) of the skid marks:

• If skid marks measure 190 feet, the speed is calculated as:
• \[v = \sqrt{24d} = \sqrt{24 \times 190} \approx 67.5 \text{ mph}\]

4. Finance

Square roots are used in financial calculations, such as determining standard deviation in statistics, which measures the volatility or risk of investments. For instance, if the variance of investment returns is 16, the standard deviation \( \sigma \) is:

• \[\sigma = \sqrt{16} = 4\]

These are just a few examples of how simplifying square root expressions is essential in solving real-world problems efficiently and accurately.

Here are some practice problems to help you master the art of simplifying square roots. Follow the steps carefully and use the rules you've learned to simplify each expression.

1. Simplify the square root of 72.
2. Simplify the square root of 200.
3. Simplify the square root of 128.
4. Simplify the square root of 45x²y³.
5. Simplify the square root of 50/2.

Solutions

1. Simplify \(\sqrt{72}\):

   \[\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}\]

2. Simplify \(\sqrt{200}\):

   \[\sqrt{200} = \sqrt{100 \times 2} = \sqrt{100} \times \sqrt{2} = 10\sqrt{2}\]

3. Simplify \(\sqrt{128}\):

   \[\sqrt{128} = \sqrt{64 \times 2} = \sqrt{64} \times \sqrt{2} = 8\sqrt{2}\]

4. Simplify \(\sqrt{45x^2y^3}\):

   \[\sqrt{45x^2y^3} = \sqrt{9 \times 5x^2y^3} = \sqrt{9} \times \sqrt{5x^2y^3} = 3\sqrt{5x^2y^3} = 3x y^{1.5} \sqrt{5}\]

5. Simplify \(\sqrt{\frac{50}{2}}\):

   \[\sqrt{\frac{50}{2}} = \sqrt{25} = 5\]

Simplifying square root expressions is a fundamental skill in mathematics that can make complex calculations more manageable. By understanding and applying the product and quotient properties of square roots, one can transform challenging radical expressions into simpler forms.

Here's a brief summary of the key points covered:

• The Product Property states that the square root of a product is equal to the product of the square roots: \( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \).
• The Quotient Property states that the square root of a quotient is equal to the quotient of the square roots: \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \).
• To simplify a square root expression:
  1. Factor the radicand (the number inside the square root) into its prime factors.
  2. Group the factors into pairs of equal factors.
  3. Move each pair of equal factors outside the square root.
  4. Simplify the expression inside the square root, if possible.

For example, to simplify \( \sqrt{72} \):

• Factor 72 into prime factors: \( 72 = 2 \times 2 \times 2 \times 3 \times 3 \).
• Group into pairs: \( (2 \times 2) \) and \( (3 \times 3) \).
• Move the pairs outside the square root: \( \sqrt{72} = 2 \times 3 \times \sqrt{2} = 6\sqrt{2} \).

Another example with a fraction: Simplify \( \sqrt{\frac{50}{2}} \):

• Use the quotient property: \( \sqrt{\frac{50}{2}} = \frac{\sqrt{50}}{\sqrt{2}} \).
• Simplify the numerator: \( \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} \).
• Thus, \( \frac{5\sqrt{2}}{\sqrt{2}} = 5 \).

By consistently practicing these steps, you can become proficient in simplifying square roots, which is essential for higher-level math and practical applications. Continue practicing with a variety of problems to strengthen your understanding and skill.