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

We are given the expression:

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

First, simplify the denominator:

\[
4\sqrt{2} - 2\sqrt{2} = (4 - 2)\sqrt{2} = 2\sqrt{2}
\]

So, the expression becomes:

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

Now, we can break this into two separate fractions:

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

Let's simplify each fraction separately:

1. For the first term:
   \[
   \frac{7\sqrt{2}}{2\sqrt{2}} = \frac{7}{2}
   \]

2. For the second term:
   \[
   \frac{3\sqrt{3}}{2\sqrt{2}} = \frac{3}{2} \cdot \frac{\sqrt{3}}{\sqrt{2}} = \frac{3}{2} \cdot \sqrt{\frac{3}{2}} = \frac{3}{2} \cdot \sqrt{\frac{3 \times 2}{2 \times 2}} = \frac{3}{2\sqrt{2}}
   \]

Therefore, the expression simplifies to:

\[
\frac{7}{2} + \frac{3}{2\sqrt{2}} \cdot \sqrt{3}
\]

Express this as $ p + q\sqrt{r} $:

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

Thus, the expression in the required form is:

\[
p = \frac{7}{2}, q = \frac{3}{2}, r = 3
\]

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

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

This is the simplest form of the given expression.

Would you like any further details or clarifications?

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Here are 8 related questions:
1. How do you rationalize the denominator of an expression?
2. What are the steps to combine like terms involving square roots?
3. How do you express a number in the form $ p + q\sqrt{r} $?
4. What does it mean for a number to be rational?
5. How do you add or subtract radical expressions?
6. How do you multiply radical expressions?
7. How can you simplify the expression $ \frac{\sqrt{a}}{\sqrt{b}} $?
8. What is the importance of simplifying expressions in algebra?

**Tip:** When dealing with radicals, always look for opportunities to simplify before performing arithmetic operations.

Learn how to simplify the expression (7√2 + 3√3) / (4√2 - 2√2) into the form p + q√r with a step-by-step solution. This problem covers rationalizing denominators, simplifying radical expressions, and expressing in the form p + q√r. Suitable for high school students.

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