Exercises 73–75 will help you prepare for the material covered in the next section. Rationalize the denominator: (7 + 4√2)/(2 - 5√2).

Verified step by step guidance

1. Identify the need to rationalize the denominator by eliminating the square root from the denominator.
2. Multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of $2-5\sqrt{2}$ is $2+5\sqrt{2}$.
3. Write the expression as $\frac{(7+4\sqrt{2})(2+5\sqrt{2})}{(2-5\sqrt{2})(2+5\sqrt{2})}$.
4. Apply the distributive property (FOIL method) to both the numerator and the denominator.
5. Simplify the expression by combining like terms and using the difference of squares formula for the denominator: $(a-b)(a+b)=a^2-b^2$.

Key Concepts

Rationalizing the Denominator

Rationalizing the denominator involves eliminating any irrational numbers from the denominator of a fraction. This is typically achieved by multiplying both the numerator and the denominator by a suitable expression that will result in a rational number in the denominator. For example, if the denominator contains a square root, multiplying by the conjugate can help achieve this.

Conjugates

The conjugate of a binomial expression is formed by changing the sign between the two terms. For instance, the conjugate of (a + b) is (a - b). When multiplying a binomial by its conjugate, the result is a difference of squares, which eliminates the square root in the denominator, making it easier to simplify the expression.

Simplifying Radicals

Simplifying radicals involves reducing a square root or other root to its simplest form. This can include factoring out perfect squares from under the radical sign or combining like terms. Understanding how to simplify radicals is essential when rationalizing denominators, as it allows for clearer and more manageable expressions.