Simplify. Assume All Variables Represent Positive Values. Do Not Use A Calculator Or Round To A Decimal. Write The Exact Simplified Form.$\[ -2 \sqrt{12x} + 9 \sqrt{75x} = \square \\]

by ADMIN 184 views

Introduction

In this article, we will simplify the given equation involving square roots. The equation is ${-2 \sqrt{12x} + 9 \sqrt{75x} = \square}$. We will assume that all variables represent positive values and will not use a calculator or round to a decimal. Our goal is to write the exact simplified form of the given equation.

Step 1: Simplify the Square Roots

To simplify the square roots, we need to find the prime factorization of the numbers inside the square roots. The prime factorization of 12 is 22⋅32^2 \cdot 3, and the prime factorization of 75 is 3⋅523 \cdot 5^2. We can rewrite the equation as:

−222⋅3⋅x+93⋅52⋅x=□{-2 \sqrt{2^2 \cdot 3 \cdot x} + 9 \sqrt{3 \cdot 5^2 \cdot x} = \square}

Step 2: Simplify the Square Roots Using the Properties of Square Roots

We can simplify the square roots using the properties of square roots. The square root of a product is equal to the product of the square roots. Therefore, we can rewrite the equation as:

−2⋅2⋅3⋅x+9⋅5⋅3⋅x=□{-2 \cdot 2 \cdot \sqrt{3 \cdot x} + 9 \cdot 5 \cdot \sqrt{3 \cdot x} = \square}

Step 3: Simplify the Equation

We can simplify the equation by combining like terms. The terms −2⋅2-2 \cdot 2 and 9⋅59 \cdot 5 can be combined as follows:

−4⋅3⋅x+45⋅3⋅x=□{-4 \cdot \sqrt{3 \cdot x} + 45 \cdot \sqrt{3 \cdot x} = \square}

Step 4: Combine Like Terms

We can combine the like terms −4⋅3⋅x-4 \cdot \sqrt{3 \cdot x} and 45⋅3⋅x45 \cdot \sqrt{3 \cdot x} as follows:

41⋅3⋅x=□{41 \cdot \sqrt{3 \cdot x} = \square}

Step 5: Write the Final Answer

The final answer is ${41 \sqrt{3x}}$. This is the exact simplified form of the given equation.

Conclusion

In this article, we simplified the given equation involving square roots. We assumed that all variables represent positive values and did not use a calculator or round to a decimal. Our goal was to write the exact simplified form of the given equation. We were able to simplify the equation by using the properties of square roots and combining like terms. The final answer is ${41 \sqrt{3x}}$.

Frequently Asked Questions

  • Q: What is the simplified form of the given equation? A: The simplified form of the given equation is ${41 \sqrt{3x}}$.
  • Q: What assumptions were made in simplifying the equation? A: We assumed that all variables represent positive values and did not use a calculator or round to a decimal.
  • Q: What properties of square roots were used in simplifying the equation? A: We used the property that the square root of a product is equal to the product of the square roots.

Related Topics

  • Simplifying equations involving square roots
  • Properties of square roots
  • Combining like terms

References

  • [1] "Simplifying Equations Involving Square Roots" by [Author]
  • [2] "Properties of Square Roots" by [Author]
  • [3] "Combining Like Terms" by [Author]

Note: The references provided are fictional and for demonstration purposes only.

-2 \sqrt{12x} + 9 \sqrt{75x} = \square }$ Q&A: Simplifying Equations Involving Square Roots

Introduction

In our previous article, we simplified the given equation involving square roots. We assumed that all variables represent positive values and did not use a calculator or round to a decimal. Our goal was to write the exact simplified form of the given equation. In this article, we will answer some frequently asked questions related to simplifying equations involving square roots.

Q: What is the simplified form of the given equation?

A: The simplified form of the given equation is ${41 \sqrt{3x}}$.

Q: What assumptions were made in simplifying the equation?

A: We assumed that all variables represent positive values and did not use a calculator or round to a decimal.

Q: What properties of square roots were used in simplifying the equation?

A: We used the property that the square root of a product is equal to the product of the square roots.

Q: How do I simplify an equation involving square roots?

A: To simplify an equation involving square roots, you need to find the prime factorization of the numbers inside the square roots. Then, you can use the properties of square roots to simplify the equation.

Q: What is the difference between simplifying an equation involving square roots and simplifying an equation involving fractions?

A: The main difference between simplifying an equation involving square roots and simplifying an equation involving fractions is that you need to use the properties of square roots to simplify the equation involving square roots. For example, you can use the property that the square root of a product is equal to the product of the square roots.

Q: Can I use a calculator to simplify an equation involving square roots?

A: No, you should not use a calculator to simplify an equation involving square roots. Instead, you should use the properties of square roots to simplify the equation.

Q: What is the importance of simplifying equations involving square roots?

A: Simplifying equations involving square roots is important because it helps you to solve the equation more easily. By simplifying the equation, you can make it easier to solve and understand.

Q: Can I simplify an equation involving square roots if the variables are negative?

A: No, you should not simplify an equation involving square roots if the variables are negative. Instead, you should assume that the variables represent positive values.

Q: What are some common mistakes to avoid when simplifying equations involving square roots?

A: Some common mistakes to avoid when simplifying equations involving square roots include:

  • Not using the properties of square roots to simplify the equation
  • Using a calculator to simplify the equation
  • Not assuming that the variables represent positive values
  • Not finding the prime factorization of the numbers inside the square roots

Conclusion

In this article, we answered some frequently asked questions related to simplifying equations involving square roots. We discussed the importance of simplifying equations involving square roots and some common mistakes to avoid. We also provided some tips on how to simplify an equation involving square roots.

Frequently Asked Questions

  • Q: What is the simplified form of the given equation? A: The simplified form of the given equation is ${41 \sqrt{3x}}$.
  • Q: What assumptions were made in simplifying the equation? A: We assumed that all variables represent positive values and did not use a calculator or round to a decimal.
  • Q: What properties of square roots were used in simplifying the equation? A: We used the property that the square root of a product is equal to the product of the square roots.

Related Topics

  • Simplifying equations involving square roots
  • Properties of square roots
  • Combining like terms

References

  • [1] "Simplifying Equations Involving Square Roots" by [Author]
  • [2] "Properties of Square Roots" by [Author]
  • [3] "Combining Like Terms" by [Author]

Note: The references provided are fictional and for demonstration purposes only.