https://r2.erweima.ai/mathbot/image/172494042096187rh3p2i.jpg

The problem in the image asks to "solve the surd" for the expression:

\[
\frac{3\sqrt{7}}{5 - \sqrt{7}}
\]

To simplify this expression, we need to rationalize the denominator. This involves multiplying both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(5 - \sqrt{7}\) is \(5 + \sqrt{7}\).

Here are the steps:

1. **Multiply the numerator and the denominator by the conjugate:**

   \[
   \frac{3\sqrt{7}}{5 - \sqrt{7}} \times \frac{5 + \sqrt{7}}{5 + \sqrt{7}}
   \]

2. **Apply the multiplication in both the numerator and the denominator:**

   The numerator becomes:
   \[
   3\sqrt{7} \times (5 + \sqrt{7}) = 15\sqrt{7} + 3 \times 7 = 15\sqrt{7} + 21
   \]

   The denominator is a difference of squares:
   \[
   (5 - \sqrt{7})(5 + \sqrt{7}) = 5^2 - (\sqrt{7})^2 = 25 - 7 = 18
   \]

3. **Simplified expression:**

   So, the simplified expression is:
   \[
   \frac{15\sqrt{7} + 21}{18}
   \]

4. **Further simplification (if needed):**

   If necessary, you can factor out common factors, but in this case, it’s already in its simplest form.

Would you like further details or have any questions about this process?

### Related Questions:

1. What is the conjugate of a binomial involving a square root?
2. How do you multiply binomials, such as \((a - b)(a + b)\)?
3. Can the expression \(\frac{15\sqrt{7} + 21}{18}\) be simplified further?
4. Why is rationalizing the denominator necessary in some cases?
5. How can you solve similar problems involving different square roots?

### Tip:
Always check if the numerator and denominator share a common factor after rationalizing the denominator, as this might allow further simplification.

Learn how to simplify the expression \(\frac{3\sqrt{7}}{5 - \sqrt{7}}\) by rationalizing the denominator. This problem involves using the conjugate of the denominator and covers key algebraic concepts, including rationalization and surds. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation