$\frac{4}{\sqrt{7} -2\sqrt{3} } = A\sqrt{7} +b\sqrt{3}$ .Find A And B

Rationalizing the Denominator and Finding the Values of a and b

In this article, we will explore the process of rationalizing the denominator of a complex fraction and use it to find the values of a and b in the given equation $\frac{4}{\sqrt{7} -2\sqrt{3} } = a\sqrt{7} +b\sqrt{3}$. Rationalizing the denominator is a crucial step in simplifying complex fractions and is a fundamental concept in algebra.

What is Rationalizing the Denominator?

Rationalizing the denominator is the process of removing the radical from the denominator of a fraction. This is done by multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of a binomial expression $a + b$ is $a - b$. In the case of a denominator with a square root, the conjugate is obtained by changing the sign of the radical.

Rationalizing the Denominator in the Given Equation

To rationalize the denominator in the given equation, we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $\sqrt{7} - 2\sqrt{3}$ is $\sqrt{7} + 2\sqrt{3}$.

\frac{4}{\sqrt{7} - 2\sqrt{3}} = \frac{4(\sqrt{7} + 2\sqrt{3})}{(\sqrt{7} - 2\sqrt{3})(\sqrt{7} + 2\sqrt{3})}

Simplifying the Expression

To simplify the expression, we need to multiply the numerator and the denominator.

\frac{4(\sqrt{7} + 2\sqrt{3})}{(\sqrt{7} - 2\sqrt{3})(\sqrt{7} + 2\sqrt{3})} = \frac{4\sqrt{7} + 8\sqrt{3}}{7 - 12}

Further Simplification

The denominator can be further simplified by combining the like terms.

\frac{4\sqrt{7} + 8\sqrt{3}}{7 - 12} = \frac{4\sqrt{7} + 8\sqrt{3}}{-5}

Equating the Expression to the Given Equation

Now that we have rationalized the denominator, we can equate the expression to the given equation.

\frac{4\sqrt{7} + 8\sqrt{3}}{-5} = a\sqrt{7} + b\sqrt{3}

Finding the Values of a and b

To find the values of a and b, we need to equate the coefficients of the square roots on both sides of the equation.

\frac{4\sqrt{7}}{-5} = a\sqrt{7}
\frac{8\sqrt{3}}{-5} = b\sqrt{3}

Solving for a and b

To solve for a and b, we can divide both sides of the equations by the coefficient of the square root.

a = \frac{4}{-5}
b = \frac{8}{-5}

In this article, we have explored the process of rationalizing the denominator of a complex fraction and used it to find the values of a and b in the given equation $\frac{4}{\sqrt{7} -2\sqrt{3} } = a\sqrt{7} +b\sqrt{3}$. We have shown that the values of a and b are $-\frac{4}{5}$ and $-\frac{8}{5}$, respectively. Rationalizing the denominator is a crucial step in simplifying complex fractions and is a fundamental concept in algebra.

The final answer is:

• a = $-\frac{4}{5}$
• b = $-\frac{8}{5}$
  Rationalizing the Denominator and Finding the Values of a and b: Q&A ====================================================================

In our previous article, we explored the process of rationalizing the denominator of a complex fraction and used it to find the values of a and b in the given equation $\frac{4}{\sqrt{7} -2\sqrt{3} } = a\sqrt{7} +b\sqrt{3}$. In this article, we will answer some of the most frequently asked questions related to rationalizing the denominator and finding the values of a and b.

Q: What is rationalizing the denominator?

A: Rationalizing the denominator is the process of removing the radical from the denominator of a fraction. This is done by multiplying both the numerator and the denominator by the conjugate of the denominator.

Q: Why do we need to rationalize the denominator?

A: We need to rationalize the denominator to simplify complex fractions and to make it easier to work with. Rationalizing the denominator is a fundamental concept in algebra and is used extensively in mathematics.

Q: How do we rationalize the denominator?

A: To rationalize the denominator, we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of a binomial expression $a + b$ is $a - b$. In the case of a denominator with a square root, the conjugate is obtained by changing the sign of the radical.

Q: What is the conjugate of a binomial expression?

A: The conjugate of a binomial expression $a + b$ is $a - b$. In the case of a denominator with a square root, the conjugate is obtained by changing the sign of the radical.

Q: How do we find the values of a and b?

A: To find the values of a and b, we need to equate the coefficients of the square roots on both sides of the equation. We can then solve for a and b by dividing both sides of the equations by the coefficient of the square root.

Q: What are the values of a and b in the given equation?

A: The values of a and b in the given equation $\frac{4}{\sqrt{7} -2\sqrt{3} } = a\sqrt{7} +b\sqrt{3}$ are $-\frac{4}{5}$ and $-\frac{8}{5}$, respectively.

Q: Why do we need to multiply both the numerator and the denominator by the conjugate of the denominator?

A: We need to multiply both the numerator and the denominator by the conjugate of the denominator to remove the radical from the denominator. This is done to simplify the fraction and to make it easier to work with.

Q: Can we rationalize the denominator of a fraction with a cube root?

A: Yes, we can rationalize the denominator of a fraction with a cube root. The process is similar to rationalizing the denominator of a fraction with a square root.

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

A: To rationalize the denominator of a fraction with a cube root, we need to multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of a binomial expression $a + b$ is $a - b$. In the case of a denominator with a cube root, the conjugate is obtained by changing the sign of the cube root.

In this article, we have answered some of the most frequently asked questions related to rationalizing the denominator and finding the values of a and b. We have shown that rationalizing the denominator is a crucial step in simplifying complex fractions and is a fundamental concept in algebra.