Express as a single fraction in simplest radical form with a rational denominator.

start fraction, 5, plus, square root of, 10, end square root, divided by, 8, plus, square root of, 10, end square root, end fraction

8+10​5+10​​


We are tasked with simplifying the expression:

\[
\frac{5 + \sqrt{10}}{8 + \sqrt{10}}
\]

We want to express this as a single fraction in simplest radical form with a rational denominator. To do this, we rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator.

### Step 1: Multiply by the conjugate of the denominator
The conjugate of \( 8 + \sqrt{10} \) is \( 8 - \sqrt{10} \). So, we multiply both the numerator and the denominator by \( 8 - \sqrt{10} \):

\[
\frac{5 + \sqrt{10}}{8 + \sqrt{10}} \times \frac{8 - \sqrt{10}}{8 - \sqrt{10}} = \frac{(5 + \sqrt{10})(8 - \sqrt{10})}{(8 + \sqrt{10})(8 - \sqrt{10})}
\]

### Step 2: Simplify the denominator
The denominator is a difference of squares:

\[
(8 + \sqrt{10})(8 - \sqrt{10}) = 8^2 - (\sqrt{10})^2 = 64 - 10 = 54
\]

### Step 3: Expand the numerator
Now, expand the product in the numerator \( (5 + \sqrt{10})(8 - \sqrt{10}) \):

\[
(5 + \sqrt{10})(8 - \sqrt{10}) = 5(8) + 5(-\sqrt{10}) + \sqrt{10}(8) + \sqrt{10}(-\sqrt{10})
\]

Simplifying each term:

\[
5(8) = 40
\]
\[
5(-\sqrt{10}) = -5\sqrt{10}
\]
\[
\sqrt{10}(8) = 8\sqrt{10}
\]
\[
\sqrt{10}(-\sqrt{10}) = -(\sqrt{10})^2 = -10
\]

Now, sum these terms:

\[
40 - 5\sqrt{10} + 8\sqrt{10} - 10 = 30 + 3\sqrt{10}
\]

### Step 4: Write the final expression
Thus, the numerator becomes \( 30 + 3\sqrt{10} \) and the denominator is \( 54 \). So, the expression simplifies to:

\[
\frac{30 + 3\sqrt{10}}{54}
\]

### Step 5: Simplify the fraction
We can factor out a 3 from the numerator:

\[
\frac{30 + 3\sqrt{10}}{54} = \frac{3(10 + \sqrt{10})}{54}
\]

Now, divide both the numerator and the denominator by 3:

\[
\frac{10 + \sqrt{10}}{18}
\]

### Final Answer:
The expression simplifies to:

\[
\frac{10 + \sqrt{10}}{18}
\]

Would you like further clarification or additional details on any step?

Here are 5 related questions to deepen understanding:
1. How do you find the conjugate of a binomial involving a square root?
2. What is the significance of rationalizing the denominator?
3. How do you simplify expressions involving square roots?
4. How does the difference of squares formula apply to binomials?
5. Can you generalize the method used here to other fractions involving square roots?

**Tip:** Always remember to factor and simplify after rationalizing the denominator—it helps make the final expression as simple as possible.

Express as a single fraction in simplest radical form with a rational denominator: \( \frac{5 + \sqrt{10}}{8 + \sqrt{10}} \)

Learn how to simplify the expression \( \frac{5 + \sqrt{10}}{8 + \sqrt{10}} \) by rationalizing the denominator. Follow a step-by-step process using the conjugate and the difference of squares formula to express the fraction in its simplest radical form. Suitable for students in Grades 9-12.

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