Math Problem Statement
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+105+10
Solution
We are tasked with simplifying the expression:
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 is . So, we multiply both the numerator and the denominator by :
Step 2: Simplify the denominator
The denominator is a difference of squares:
Step 3: Expand the numerator
Now, expand the product in the numerator :
Simplifying each term:
Now, sum these terms:
Step 4: Write the final expression
Thus, the numerator becomes and the denominator is . So, the expression simplifies to:
Step 5: Simplify the fraction
We can factor out a 3 from the numerator:
Now, divide both the numerator and the denominator by 3:
Final Answer:
The expression simplifies to:
Would you like further clarification or additional details on any step?
Here are 5 related questions to deepen understanding:
- How do you find the conjugate of a binomial involving a square root?
- What is the significance of rationalizing the denominator?
- How do you simplify expressions involving square roots?
- How does the difference of squares formula apply to binomials?
- 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.
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Math Problem Analysis
Mathematical Concepts
Rationalizing the Denominator
Algebraic Fractions
Square Roots
Formulas
Difference of Squares: (a + b)(a - b) = a^2 - b^2
Theorems
Conjugate Multiplication for Rationalizing Denominators
Suitable Grade Level
Grades 9-12
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