Express $$ \frac{7\sqrt2+3\sqrt3}{4\sqrt2-2\sqrt3} $$,in term of p+q$$ \sqrt{r}\,\text{where }p,q\text{ abd,r,are,rational number} $$

We need to simplify the expression:

\[
\frac{7\sqrt{2} + 3\sqrt{3}}{4\sqrt{2} - 2\sqrt{3}}
\]

To express this in the form $ p + q\sqrt{r} $, where $ p $, $ q $, and $ r $ are rational numbers, we need to rationalize the denominator. This can be done by multiplying both the numerator and the denominator by the conjugate of the denominator.

The conjugate of $ 4\sqrt{2} - 2\sqrt{3} $ is $ 4\sqrt{2} + 2\sqrt{3} $. So, we multiply the numerator and denominator by this conjugate:

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

Now, let's simplify the denominator first:

\[
(4\sqrt{2} - 2\sqrt{3})(4\sqrt{2} + 2\sqrt{3}) = (4\sqrt{2})^2 - (2\sqrt{3})^2
\]
\[
= 16 \times 2 - 4 \times 3 = 32 - 12 = 20
\]

So, the denominator simplifies to 20.

Now, let's expand the numerator:

\[
(7\sqrt{2})(4\sqrt{2}) + (7\sqrt{2})(2\sqrt{3}) + (3\sqrt{3})(4\sqrt{2}) + (3\sqrt{3})(2\sqrt{3})
\]

Simplifying each term:

1. $ (7\sqrt{2})(4\sqrt{2}) = 28 \times 2 = 56 $
2. $ (7\sqrt{2})(2\sqrt{3}) = 14\sqrt{6} $
3. $ (3\sqrt{3})(4\sqrt{2}) = 12\sqrt{6} $
4. $ (3\sqrt{3})(2\sqrt{3}) = 6 \times 3 = 18 $

Adding these together:

\[
56 + 18 + 14\sqrt{6} + 12\sqrt{6} = 74 + 26\sqrt{6}
\]

Thus, the expression simplifies to:

\[
\frac{74 + 26\sqrt{6}}{20}
\]

We can now simplify by dividing both terms in the numerator by 20:

\[
\frac{74}{20} + \frac{26\sqrt{6}}{20} = \frac{37}{10} + \frac{13\sqrt{6}}{10}
\]

So, the expression in the form $ p + q\sqrt{r} $ is:

\[
p = \frac{37}{10}, \, q = \frac{13}{10}, \, r = 6
\]

The final expression is:

\[
\frac{37}{10} + \frac{13}{10}\sqrt{6}
\]

Would you like any further details or clarifications?

---

Here are 8 related questions:
1. How do you find the conjugate of a binomial involving square roots?
2. What is the process for rationalizing a denominator?
3. How do you multiply binomials involving radicals?
4. What are the steps to simplify a radical expression?
5. How do you combine like terms involving square roots?
6. How can you simplify the expression $ \sqrt{a} \times \sqrt{b} $?
7. What is the difference between rational and irrational numbers?
8. How do you simplify expressions involving both rational and irrational components?

**Tip:** Always remember to look for opportunities to factor or rationalize expressions involving radicals before simplifying.

Learn how to simplify the expression (7sqrt{2} + 3sqrt{3}) / (4sqrt{2} - 2sqrt{3}) using rationalization and simplification of radical expressions. This problem involves rationalizing the denominator by multiplying with the conjugate and simplifying the resulting expression. Suitable for high school students studying algebra and radicals.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation