Use The Properties Of Radicals To Simplify The Expression.$\sqrt{2} \cdot \sqrt{72} = $ □ \square □

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Understanding Radicals and Their Properties

Radicals, also known as roots, are mathematical expressions that involve the extraction of a root of a number. The most common radical is the square root, denoted by the symbol \sqrt{ }. In this article, we will focus on simplifying radical expressions using the properties of radicals.

The Properties of Radicals

Before we dive into simplifying radical expressions, it's essential to understand the properties of radicals. The following are the key properties of radicals:

  • Product Property: a⋅b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}
  • Quotient Property: ab=ab\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}
  • Power Property: (a)n=an2(\sqrt{a})^n = a^{\frac{n}{2}}

Simplifying Radical Expressions

Now that we have a good understanding of the properties of radicals, let's apply them to simplify the given expression: 2⋅72\sqrt{2} \cdot \sqrt{72}.

Step 1: Break Down the Radicals

To simplify the expression, we need to break down the radicals into their prime factors. The prime factorization of 72 is 23⋅322^3 \cdot 3^2. Therefore, we can rewrite the expression as:

2⋅23⋅32\sqrt{2} \cdot \sqrt{2^3 \cdot 3^2}

Step 2: Apply the Product Property

Now that we have broken down the radicals, we can apply the product property to simplify the expression. According to the product property, we can combine the two radicals into a single radical:

2⋅23⋅32=2⋅23⋅32\sqrt{2} \cdot \sqrt{2^3 \cdot 3^2} = \sqrt{2 \cdot 2^3 \cdot 3^2}

Step 3: Simplify the Expression

Now that we have combined the two radicals, we can simplify the expression by combining like terms. The expression can be rewritten as:

2⋅23⋅32=24⋅32\sqrt{2 \cdot 2^3 \cdot 3^2} = \sqrt{2^4 \cdot 3^2}

Step 4: Apply the Power Property

Finally, we can apply the power property to simplify the expression. According to the power property, we can rewrite the radical as:

24⋅32=(22⋅3)12\sqrt{2^4 \cdot 3^2} = (2^2 \cdot 3)^{\frac{1}{2}}

Step 5: Simplify the Expression

Now that we have applied the power property, we can simplify the expression by evaluating the exponent:

(22⋅3)12=(12)12(2^2 \cdot 3)^{\frac{1}{2}} = (12)^{\frac{1}{2}}

Step 6: Evaluate the Square Root

Finally, we can evaluate the square root to get the final answer:

(12)12=23(12)^{\frac{1}{2}} = \boxed{2\sqrt{3}}

Conclusion

In this article, we have learned how to simplify radical expressions using the properties of radicals. We have applied the product property, quotient property, and power property to simplify the expression 2⋅72\sqrt{2} \cdot \sqrt{72}. The final answer is 23\boxed{2\sqrt{3}}.

Common Mistakes to Avoid

When simplifying radical expressions, there are several common mistakes to avoid:

  • Not breaking down the radicals into their prime factors
  • Not applying the product property
  • Not simplifying the expression by combining like terms
  • Not applying the power property

Tips and Tricks

Here are some tips and tricks to help you simplify radical expressions:

  • Always break down the radicals into their prime factors
  • Apply the product property to combine the radicals
  • Simplify the expression by combining like terms
  • Apply the power property to rewrite the radical

Practice Problems

Here are some practice problems to help you practice simplifying radical expressions:

  • $\sqrt{3} \cdot \sqrt{12} = $
  • $\frac{\sqrt{2}}{\sqrt{3}} = $
  • $(\sqrt{2})^3 = $

Solutions

Here are the solutions to the practice problems:

  • 3⋅12=23\sqrt{3} \cdot \sqrt{12} = \boxed{2\sqrt{3}}
  • 23=23\frac{\sqrt{2}}{\sqrt{3}} = \boxed{\frac{\sqrt{2}}{\sqrt{3}}}
  • (2)3=22(\sqrt{2})^3 = \boxed{2\sqrt{2}}

Conclusion

In this article, we have learned how to simplify radical expressions using the properties of radicals. We have applied the product property, quotient property, and power property to simplify the expression 2⋅72\sqrt{2} \cdot \sqrt{72}. The final answer is 23\boxed{2\sqrt{3}}. We have also provided some tips and tricks to help you simplify radical expressions and some practice problems to help you practice.

Q: What is the product property of radicals?

A: The product property of radicals states that a⋅b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}. This means that we can combine two radicals into a single radical by multiplying their radicands.

Q: How do I simplify a radical expression using the product property?

A: To simplify a radical expression using the product property, you need to break down the radicals into their prime factors and then combine them. For example, if you have the expression 2⋅72\sqrt{2} \cdot \sqrt{72}, you can break down the radicals into their prime factors as follows:

2⋅23⋅32\sqrt{2} \cdot \sqrt{2^3 \cdot 3^2}

Then, you can combine the two radicals into a single radical using the product property:

2⋅23⋅32=2⋅23⋅32\sqrt{2} \cdot \sqrt{2^3 \cdot 3^2} = \sqrt{2 \cdot 2^3 \cdot 3^2}

Q: What is the quotient property of radicals?

A: The quotient property of radicals states that ab=ab\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}. This means that we can divide two radicals by dividing their radicands.

Q: How do I simplify a radical expression using the quotient property?

A: To simplify a radical expression using the quotient property, you need to divide the radicands of the two radicals. For example, if you have the expression 23\frac{\sqrt{2}}{\sqrt{3}}, you can simplify it using the quotient property as follows:

23=23\frac{\sqrt{2}}{\sqrt{3}} = \sqrt{\frac{2}{3}}

Q: What is the power property of radicals?

A: The power property of radicals states that (a)n=an2(\sqrt{a})^n = a^{\frac{n}{2}}. This means that we can raise a radical to a power by raising its radicand to half the power.

Q: How do I simplify a radical expression using the power property?

A: To simplify a radical expression using the power property, you need to raise the radicand to half the power. For example, if you have the expression (2)3(\sqrt{2})^3, you can simplify it using the power property as follows:

(2)3=232(\sqrt{2})^3 = 2^{\frac{3}{2}}

Q: Can I simplify a radical expression if it has a negative exponent?

A: Yes, you can simplify a radical expression if it has a negative exponent. To do this, you need to apply the power property and then simplify the resulting expression. For example, if you have the expression (2)−3(\sqrt{2})^{-3}, you can simplify it as follows:

(2)−3=2−32(\sqrt{2})^{-3} = 2^{-\frac{3}{2}}

Q: Can I simplify a radical expression if it has a fractional exponent?

A: Yes, you can simplify a radical expression if it has a fractional exponent. To do this, you need to apply the power property and then simplify the resulting expression. For example, if you have the expression (2)13(\sqrt{2})^{\frac{1}{3}}, you can simplify it as follows:

(2)13=216(\sqrt{2})^{\frac{1}{3}} = 2^{\frac{1}{6}}

Q: How do I know if a radical expression can be simplified?

A: To determine if a radical expression can be simplified, you need to check if the radicand can be broken down into its prime factors. If the radicand can be broken down into its prime factors, then the radical expression can be simplified using the product property, quotient property, or power property.

Q: What are some common mistakes to avoid when simplifying radical expressions?

A: Some common mistakes to avoid when simplifying radical expressions include:

  • Not breaking down the radicals into their prime factors
  • Not applying the product property
  • Not simplifying the expression by combining like terms
  • Not applying the power property

Q: How can I practice simplifying radical expressions?

A: You can practice simplifying radical expressions by working through practice problems and exercises. You can also try simplifying different types of radical expressions, such as those with negative exponents or fractional exponents.

Q: What are some real-world applications of simplifying radical expressions?

A: Simplifying radical expressions has many real-world applications, including:

  • Engineering: Simplifying radical expressions is used in engineering to solve problems involving geometry and trigonometry.
  • Physics: Simplifying radical expressions is used in physics to solve problems involving motion and energy.
  • Computer Science: Simplifying radical expressions is used in computer science to solve problems involving algorithms and data structures.

Q: Can I use a calculator to simplify radical expressions?

A: Yes, you can use a calculator to simplify radical expressions. However, it's always a good idea to check your work by hand to make sure that the calculator is giving you the correct answer.

Q: How can I check my work when simplifying radical expressions?

A: You can check your work when simplifying radical expressions by:

  • Re-checking the radicand: Make sure that the radicand can be broken down into its prime factors.
  • Re-checking the exponent: Make sure that the exponent is correct.
  • Re-checking the simplified expression: Make sure that the simplified expression is correct.

Q: What are some common mistakes to avoid when checking your work?

A: Some common mistakes to avoid when checking your work include:

  • Not re-checking the radicand
  • Not re-checking the exponent
  • Not re-checking the simplified expression

Q: How can I improve my skills when simplifying radical expressions?

A: You can improve your skills when simplifying radical expressions by:

  • Practicing regularly: Practice simplifying radical expressions regularly to build your skills.
  • Reviewing concepts: Review the concepts of radicals and exponents to make sure that you understand them.
  • Seeking help: Seek help from a teacher or tutor if you are having trouble simplifying radical expressions.