Find The Value Of Each Expression.C) 162 ⋅ 2 \sqrt{162} \cdot \sqrt{2} 162 ​ ⋅ 2 ​ Answer:

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Introduction

Radical expressions are an essential part of mathematics, and simplifying them is a crucial skill to master. In this article, we will focus on simplifying the expression 162⋅2\sqrt{162} \cdot \sqrt{2}, and we will explore the steps involved in finding the value of this expression.

Understanding the Expression

The given expression is 162⋅2\sqrt{162} \cdot \sqrt{2}. To simplify this expression, we need to understand the properties of square roots. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16.

Breaking Down the Expression

To simplify the expression 162⋅2\sqrt{162} \cdot \sqrt{2}, we need to break it down into smaller parts. We can start by finding the prime factorization of the numbers involved.

Prime Factorization

The prime factorization of 162 is 2⋅342 \cdot 3^4, and the prime factorization of 2 is simply 2. Now that we have the prime factorization of both numbers, we can rewrite the expression as 2⋅34⋅2\sqrt{2 \cdot 3^4} \cdot \sqrt{2}.

Using the Properties of Square Roots

Now that we have rewritten the expression, we can use the properties of square roots to simplify it. One of the properties of square roots is that the square root of a product is equal to the product of the square roots. Using this property, we can rewrite the expression as 2⋅34⋅2\sqrt{2 \cdot 3^4 \cdot 2}.

Simplifying the Expression

Now that we have rewritten the expression, we can simplify it by combining the numbers under the square root. We can start by combining the 2's, which gives us 22⋅34\sqrt{2^2 \cdot 3^4}.

Using the Property of Square Roots Again

Now that we have simplified the expression, we can use the property of square roots again to rewrite it as (22)⋅(34)\sqrt{(2^2) \cdot (3^4)}.

Simplifying Further

Now that we have rewritten the expression, we can simplify it further by evaluating the expressions under the square root. We can start by evaluating 222^2, which equals 4. We can then evaluate 343^4, which equals 81.

Finding the Final Answer

Now that we have simplified the expression, we can find the final answer. We can start by multiplying the numbers under the square root, which gives us 4⋅81\sqrt{4 \cdot 81}. We can then simplify this expression by evaluating the product under the square root, which gives us 324\sqrt{324}.

Evaluating the Square Root

Now that we have simplified the expression, we can evaluate the square root. We can start by finding the prime factorization of 324, which is 22⋅342^2 \cdot 3^4. We can then use this prime factorization to rewrite the expression as 22⋅34\sqrt{2^2 \cdot 3^4}.

Finding the Final Answer

Now that we have evaluated the square root, we can find the final answer. We can start by simplifying the expression under the square root, which gives us 22⋅34\sqrt{2^2 \cdot 3^4}. We can then evaluate this expression by multiplying the numbers under the square root, which gives us 18\boxed{18}.

Conclusion

In this article, we have simplified the expression 162⋅2\sqrt{162} \cdot \sqrt{2} by breaking it down into smaller parts and using the properties of square roots. We have shown that the final answer is 18\boxed{18}, and we have provided a step-by-step guide to help readers understand the process involved in simplifying radical expressions.

Additional Tips and Tricks

  • When simplifying radical expressions, it is essential to understand the properties of square roots.
  • Breaking down the expression into smaller parts can help make it easier to simplify.
  • Using the properties of square roots can help simplify the expression and find the final answer.

Common Mistakes to Avoid

  • Not understanding the properties of square roots can lead to incorrect simplifications.
  • Not breaking down the expression into smaller parts can make it difficult to simplify.
  • Not using the properties of square roots can lead to incorrect final answers.

Real-World Applications

Simplifying radical expressions is an essential skill in mathematics, and it has many real-world applications. For example, in engineering, simplifying radical expressions can help design and build structures that are safe and efficient. In finance, simplifying radical expressions can help calculate interest rates and investments. In science, simplifying radical expressions can help understand complex phenomena and make predictions.

Final Thoughts

Introduction

In our previous article, we explored the process of simplifying radical expressions, including the expression 162⋅2\sqrt{162} \cdot \sqrt{2}. We provided a step-by-step guide to help readers understand the process involved in simplifying radical expressions. In this article, we will answer some of the most frequently asked questions about simplifying radical expressions.

Q: What is a radical expression?

A: A radical expression is an expression that contains a square root or a cube root. For example, 16\sqrt{16} and 273\sqrt[3]{27} are both radical expressions.

Q: How do I simplify a radical expression?

A: To simplify a radical expression, you need to understand the properties of square roots and break down the expression into smaller parts. You can start by finding the prime factorization of the numbers involved and then use the properties of square roots to simplify the expression.

Q: What are the properties of square roots?

A: The properties of square roots include:

  • The square root of a product is equal to the product of the square roots.
  • The square root of a quotient is equal to the quotient of the square roots.
  • The square root of a number can be simplified by finding the prime factorization of the number.

Q: How do I find the prime factorization of a number?

A: To find the prime factorization of a number, you need to break down the number into its prime factors. For example, the prime factorization of 12 is 22⋅32^2 \cdot 3.

Q: What is the difference between a square root and a cube root?

A: A square root is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4, because 4 multiplied by 4 equals 16. A cube root is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 27 is 3, because 3 multiplied by 3 multiplied by 3 equals 27.

Q: How do I simplify a radical expression with a cube root?

A: To simplify a radical expression with a cube root, you need to understand the properties of cube roots and break down the expression into smaller parts. You can start by finding the prime factorization of the numbers involved and then use the properties of cube roots to simplify the expression.

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

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

  • Not understanding the properties of square roots and cube roots.
  • Not breaking down the expression into smaller parts.
  • Not using the properties of square roots and cube roots to simplify the expression.

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 numbers involved have any common factors. If they do, you can simplify the expression by canceling out the common factors.

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

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

  • Designing and building structures that are safe and efficient.
  • Calculating interest rates and investments.
  • Understanding complex phenomena and making predictions.

Conclusion

Simplifying radical expressions is a crucial skill in mathematics, and it has many real-world applications. By understanding the properties of square roots and breaking down the expression into smaller parts, we can simplify radical expressions and find the final answer. We hope that this article has provided a helpful guide to simplifying radical expressions and has inspired readers to explore the world of mathematics.

Additional Resources

  • For more information on simplifying radical expressions, check out our previous article on the topic.
  • For practice problems and exercises, try our online math resources.
  • For more advanced topics in mathematics, check out our articles on calculus and differential equations.