Simplify The Expression: X = 14 3 ⋅ 2 X = 14 \sqrt{3} \cdot 2 X = 14 3 ​ ⋅ 2

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently and accurately. When we simplify an expression, we combine like terms and eliminate any unnecessary components. In this article, we will focus on simplifying the given expression: x=143⋅2x = 14 \sqrt{3} \cdot 2. We will break down the steps involved in simplifying this expression and provide a clear explanation of each step.

Understanding the Expression

The given expression is x=143⋅2x = 14 \sqrt{3} \cdot 2. To simplify this expression, we need to understand the properties of radicals and how they interact with multiplication. The expression involves a square root of 3, which is a radical. When we multiply a radical by a number, we can multiply the number inside the radical by the number outside the radical.

Simplifying the Expression

To simplify the expression, we can start by multiplying the number outside the radical, which is 14, by the number inside the radical, which is 2. This gives us:

x=14⋅2⋅3x = 14 \cdot 2 \cdot \sqrt{3}

Next, we can multiply the numbers outside the radical, which gives us:

x=28⋅3x = 28 \cdot \sqrt{3}

Properties of Radicals

When we multiply a radical by a number, we can multiply the number inside the radical by the number outside the radical. This property is known as the product rule for radicals. The product rule states that:

a⋅b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}

Using this property, we can simplify the expression further:

x=28⋅3x = 28 \cdot \sqrt{3}

x=(28)2⋅3x = \sqrt{(28)^2 \cdot 3}

x=784⋅3x = \sqrt{784 \cdot 3}

x=2352x = \sqrt{2352}

Final Simplification

The expression x=2352x = \sqrt{2352} can be simplified further by finding the prime factorization of 2352. The prime factorization of 2352 is:

2352=24⋅32⋅722352 = 2^4 \cdot 3^2 \cdot 7^2

Using this prime factorization, we can simplify the expression:

x=24⋅32⋅72x = \sqrt{2^4 \cdot 3^2 \cdot 7^2}

x=22⋅3⋅7x = 2^2 \cdot 3 \cdot 7

x=4⋅3⋅7x = 4 \cdot 3 \cdot 7

x=84x = 84

Conclusion

In this article, we simplified the expression x=143⋅2x = 14 \sqrt{3} \cdot 2 by applying the properties of radicals and multiplying the numbers outside and inside the radical. We used the product rule for radicals to simplify the expression and found the prime factorization of 2352 to further simplify the expression. The final simplified expression is x=84x = 84.

Frequently Asked Questions

  • What is the product rule for radicals? The product rule for radicals states that a⋅b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}.
  • How do I simplify an expression with a radical? To simplify an expression with a radical, you can multiply the number inside the radical by the number outside the radical.
  • What is the prime factorization of 2352? The prime factorization of 2352 is 24⋅32⋅722^4 \cdot 3^2 \cdot 7^2.

Additional Resources

  • Khan Academy: Simplifying Radicals
  • Mathway: Simplifying Expressions with Radicals
  • Wolfram Alpha: Simplifying Expressions with Radicals

Introduction

In our previous article, we simplified the expression x=143⋅2x = 14 \sqrt{3} \cdot 2 by applying the properties of radicals and multiplying the numbers outside and inside the radical. We used the product rule for radicals to simplify the expression and found the prime factorization of 2352 to further simplify the expression. In this article, we will answer some frequently asked questions related to simplifying expressions with radicals.

Q&A

Q: What is the product rule for radicals?

A: The product rule for radicals states that a⋅b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}. This means that when we multiply two radicals, we can multiply the numbers inside the radicals and take the square root of the product.

Q: How do I simplify an expression with a radical?

A: To simplify an expression with a radical, you can multiply the number inside the radical by the number outside the radical. For example, if we have the expression x=143⋅2x = 14 \sqrt{3} \cdot 2, we can multiply the number outside the radical, which is 14, by the number inside the radical, which is 2.

Q: What is the difference between a radical and a rational number?

A: A radical is a number that can be expressed as the square root of a number, such as 3\sqrt{3} or 4\sqrt{4}. A rational number is a number that can be expressed as the ratio of two integers, such as 3/4 or 2/3.

Q: Can I simplify an expression with a radical by multiplying the numbers inside the radical?

A: Yes, you can simplify an expression with a radical by multiplying the numbers inside the radical. For example, if we have the expression x=12x = \sqrt{12}, we can multiply the numbers inside the radical to get x=4⋅3=23x = \sqrt{4 \cdot 3} = 2 \sqrt{3}.

Q: How do I simplify an expression with a radical that has a coefficient?

A: To simplify an expression with a radical that has a coefficient, you can multiply the coefficient by the number inside the radical. For example, if we have the expression x=34x = 3 \sqrt{4}, we can multiply the coefficient, which is 3, by the number inside the radical, which is 4.

Q: Can I simplify an expression with a radical by using the prime factorization of the number inside the radical?

A: Yes, you can simplify an expression with a radical by using the prime factorization of the number inside the radical. For example, if we have the expression x=24x = \sqrt{24}, we can find the prime factorization of 24, which is 23⋅32^3 \cdot 3. We can then simplify the expression by taking the square root of the prime factorization.

Q: How do I simplify an expression with a radical that has a negative number inside the radical?

A: To simplify an expression with a radical that has a negative number inside the radical, you can multiply the number inside the radical by the number outside the radical. For example, if we have the expression x=−4x = \sqrt{-4}, we can multiply the number inside the radical, which is -4, by the number outside the radical, which is 2.

Conclusion

In this article, we answered some frequently asked questions related to simplifying expressions with radicals. We discussed the product rule for radicals, how to simplify expressions with radicals, and how to simplify expressions with radicals that have coefficients or negative numbers inside the radical. We also discussed how to simplify expressions with radicals by using the prime factorization of the number inside the radical.

Frequently Asked Questions

  • What is the product rule for radicals?
  • How do I simplify an expression with a radical?
  • What is the difference between a radical and a rational number?
  • Can I simplify an expression with a radical by multiplying the numbers inside the radical?
  • How do I simplify an expression with a radical that has a coefficient?
  • Can I simplify an expression with a radical by using the prime factorization of the number inside the radical?
  • How do I simplify an expression with a radical that has a negative number inside the radical?

Additional Resources

  • Khan Academy: Simplifying Radicals
  • Mathway: Simplifying Expressions with Radicals
  • Wolfram Alpha: Simplifying Expressions with Radicals