Use The Equation $y=\sqrt[3]{27x-54}+5$. Which Is An Equivalent Equation Of The Form $y=a \sqrt{x-h}+k$?A. $y=-27 \sqrt[3]{x+2}+5$B. $y=-3 \sqrt[3]{x+2}+5$C. $y=3 \sqrt[3]{x-2}+5$D. $y=27

by ADMIN 188 views

Introduction

Radical expressions are a fundamental concept in mathematics, and rationalizing the denominator is a crucial skill to master when working with these expressions. In this article, we will explore the concept of rationalizing the denominator and provide a step-by-step guide on how to simplify radical expressions.

What is Rationalizing the Denominator?

Rationalizing the denominator is the process of eliminating any radicals from the denominator of a fraction. This is done by multiplying both the numerator and the denominator by a cleverly chosen value that will eliminate the radical in the denominator.

Why is Rationalizing the Denominator Important?

Rationalizing the denominator is an essential skill in mathematics because it allows us to simplify complex radical expressions and make them easier to work with. By eliminating the radical in the denominator, we can perform operations such as addition, subtraction, multiplication, and division with greater ease.

Step-by-Step Guide to Rationalizing the Denominator

To rationalize the denominator, follow these steps:

  1. Identify the radical in the denominator: The first step is to identify the radical in the denominator. In the equation y=27x−543+5y=\sqrt[3]{27x-54}+5, the radical is 27x−543\sqrt[3]{27x-54}.
  2. Determine the value to multiply by: To eliminate the radical in the denominator, we need to multiply both the numerator and the denominator by a value that will eliminate the radical. In this case, we need to multiply by 273\sqrt[3]{27}.
  3. Multiply the numerator and the denominator: Multiply both the numerator and the denominator by 273\sqrt[3]{27}.
  4. Simplify the expression: Simplify the expression by combining like terms and eliminating any radicals in the denominator.

Example: Rationalizing the Denominator in the Equation y=27x−543+5y=\sqrt[3]{27x-54}+5

Let's apply the steps above to the equation y=27x−543+5y=\sqrt[3]{27x-54}+5.

  1. Identify the radical in the denominator: The radical in the denominator is 27x−543\sqrt[3]{27x-54}.
  2. Determine the value to multiply by: To eliminate the radical in the denominator, we need to multiply both the numerator and the denominator by 273\sqrt[3]{27}.
  3. Multiply the numerator and the denominator: Multiply both the numerator and the denominator by 273\sqrt[3]{27}.
  4. Simplify the expression: Simplify the expression by combining like terms and eliminating any radicals in the denominator.

Solving for the Equivalent Equation

To solve for the equivalent equation of the form y=ax−h+ky=a \sqrt{x-h}+k, we need to manipulate the equation y=27x−543+5y=\sqrt[3]{27x-54}+5.

First, we can rewrite the equation as y=27(x−2)3+5y=\sqrt[3]{27(x-2)}+5.

Next, we can factor out the 3 from the expression inside the cube root: y=33(x−2)3+5y=\sqrt[3]{3^3(x-2)}+5.

Now, we can simplify the expression by combining like terms: y=3x−23+5y=3\sqrt[3]{x-2}+5.

Conclusion

In this article, we explored the concept of rationalizing the denominator and provided a step-by-step guide on how to simplify radical expressions. We applied the steps to the equation y=27x−543+5y=\sqrt[3]{27x-54}+5 and solved for the equivalent equation of the form y=ax−h+ky=a \sqrt{x-h}+k. By mastering the skill of rationalizing the denominator, we can simplify complex radical expressions and make them easier to work with.

Final Answer

The final answer is: C.y=3x−23+5\boxed{C. y=3 \sqrt[3]{x-2}+5}

Introduction

Radical expressions are a fundamental concept in mathematics, and rationalizing the denominator is a crucial skill to master when working with these expressions. In this article, we will explore the concept of rationalizing the denominator and provide a Q&A guide to help you understand and simplify radical expressions.

Q&A: Rationalizing the Denominator

Q: What is rationalizing the denominator?

A: Rationalizing the denominator is the process of eliminating any radicals from the denominator of a fraction. This is done by multiplying both the numerator and the denominator by a cleverly chosen value that will eliminate the radical in the denominator.

Q: Why is rationalizing the denominator important?

A: Rationalizing the denominator is an essential skill in mathematics because it allows us to simplify complex radical expressions and make them easier to work with. By eliminating the radical in the denominator, we can perform operations such as addition, subtraction, multiplication, and division with greater ease.

Q: How do I rationalize the denominator?

A: To rationalize the denominator, follow these steps:

  1. Identify the radical in the denominator: The first step is to identify the radical in the denominator.
  2. Determine the value to multiply by: To eliminate the radical in the denominator, we need to multiply both the numerator and the denominator by a value that will eliminate the radical.
  3. Multiply the numerator and the denominator: Multiply both the numerator and the denominator by the value determined in step 2.
  4. Simplify the expression: Simplify the expression by combining like terms and eliminating any radicals in the denominator.

Q: What is the difference between rationalizing the denominator and simplifying a radical expression?

A: Rationalizing the denominator is the process of eliminating any radicals from the denominator of a fraction, while simplifying a radical expression involves rewriting the expression in its simplest form.

Q: Can I rationalize the denominator of a fraction with a negative exponent?

A: Yes, you can rationalize the denominator of a fraction with a negative exponent. However, you will need to follow the same steps as before, but with a negative exponent.

Q: How do I rationalize the denominator of a fraction with a cube root?

A: To rationalize the denominator of a fraction with a cube root, you will need to multiply both the numerator and the denominator by the cube root of the value that will eliminate the cube root in the denominator.

Q: Can I rationalize the denominator of a fraction with a square root?

A: Yes, you can rationalize the denominator of a fraction with a square root. However, you will need to follow the same steps as before, but with a square root.

Example: Rationalizing the Denominator of a Fraction with a Cube Root

Let's say we have the fraction x−23x+23\frac{\sqrt[3]{x-2}}{\sqrt[3]{x+2}}. To rationalize the denominator, we need to multiply both the numerator and the denominator by the cube root of the value that will eliminate the cube root in the denominator.

Solving for the Rationalized Denominator

To solve for the rationalized denominator, we can multiply both the numerator and the denominator by x+23\sqrt[3]{x+2}.

Simplifying the Expression

Simplifying the expression, we get x−23x+23x+23x+23\frac{\sqrt[3]{x-2}\sqrt[3]{x+2}}{\sqrt[3]{x+2}\sqrt[3]{x+2}}.

Final Answer

The final answer is: x2−43x+2\boxed{\frac{\sqrt[3]{x^2-4}}{x+2}}

Conclusion

In this article, we explored the concept of rationalizing the denominator and provided a Q&A guide to help you understand and simplify radical expressions. By mastering the skill of rationalizing the denominator, you can simplify complex radical expressions and make them easier to work with.

Final Tips

  • Always identify the radical in the denominator before attempting to rationalize it.
  • Determine the value to multiply by carefully to ensure that it will eliminate the radical in the denominator.
  • Simplify the expression by combining like terms and eliminating any radicals in the denominator.
  • Practice, practice, practice! The more you practice rationalizing the denominator, the more comfortable you will become with the process.