How can I rationalize a function

Question

ScanSolve AI Tutor · Accepted Answer

Rationalizing a function typically means eliminating radical expressions (like square roots) from the denominator or numerator of a fraction. This is often done to simplify expressions or to prepare them for further calculations, such as finding limits.

Here's how to rationalize a function:

Identify the radical expression you want to eliminate.
This is usually in the denominator.
Determine the appropriate factor to multiply by.
- If the radical is a single term, like $\sqrt{a}$ , multiply the numerator and denominator by $\frac{\sqrt{a}}{\sqrt{a}}$ .
- If the radical is part of a binomial expression, like $a + \sqrt{b}$ or $\sqrt{a} + \sqrt{b}$ , multiply the numerator and denominator by its conjugate. The conjugate of $A+B$ is $A-B$ , and vice versa. This uses the difference of squares formula: $(A+B)(A-B) = A^2 - B^2$ .
Multiply the numerator and denominator by the chosen factor and simplify the expression.
Example: Rationalize the denominator of $\frac{1}{1+\sqrt{3}}$ .
The radical expression in the denominator is $1+\sqrt{3}$ .
The denominator is a binomial with a radical.
Its conjugate is $1-\sqrt{3}$ . We multiply the fraction by $\frac{1-\sqrt{3}}{1-\sqrt{3}}$ .
$\frac{1}{1+\sqrt{3}} \times \frac{1-\sqrt{3}}{1-\sqrt{3}}$
Perform the multiplication and simplify.
$\frac{1(1-\sqrt{3})}{(1+\sqrt{3})(1-\sqrt{3})} = \frac{1-\sqrt{3}}{1^2 - (\sqrt{3})^2} = \frac{1-\sqrt{3}}{1 - 3} = \frac{1-\sqrt{3}}{-2}$
The rationalized function is:
$\frac{\sqrt{3}-1}{2}$

How can I rationalize a function

Answer

Step-by-step solution

Identify the radical expression you want to eliminate.

Determine the appropriate factor to multiply by.

Multiply the numerator and denominator by the chosen factor and simplify the expression.

The radical expression in the denominator is $1+\sqrt{3}$ .

The denominator is a binomial with a radical.

Perform the multiplication and simplify.

Try one yourself

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More Computer Science Questions

How can I rationalize a function

Answer

Step-by-step solution

Identify the radical expression you want to eliminate.

Determine the appropriate factor to multiply by.

Multiply the numerator and denominator by the chosen factor and simplify the expression.

The radical expression in the denominator is 1+31+\sqrt{3}1+3​.

The denominator is a binomial with a radical.

Perform the multiplication and simplify.

Try one yourself

Still stuck on this one?

More Computer Science Questions

The radical expression in the denominator is $1+\sqrt{3}$ .