Write the Fraction in Simplest Form ( square root of 2+ square root of 5)/( square root of 2- square root of 5)
This problem is asking you to simplify a fraction where both the numerator (top part) and the denominator (bottom part) contain radical expressions (in this case, square roots). Specifically, the fraction is composed of the sum of the square roots of 2 and 5 in the numerator and the difference of the same square roots in the denominator. The challenge involves simplifying this complex fraction into its simplest form, which typically means rationalizing the denominator by eliminating the radical expressions, so that no square roots appear in the denominator of the final simplified result.
$\frac{\sqrt{2} + \sqrt{5}}{\sqrt{2} - \sqrt{5}}$
Answer
Solution:
Step 1:
Multiply the given fraction by a fraction that is equivalent to 1, which has the conjugate of the denominator as both the numerator and the denominator: $\frac{\sqrt{2} + \sqrt{5}}{\sqrt{2} - \sqrt{5}} \cdot \frac{\sqrt{2} + \sqrt{5}}{\sqrt{2} + \sqrt{5}}$.
Step 2:
Apply the conjugate multiplication to the fraction: $\frac{(\sqrt{2} + \sqrt{5})(\sqrt{2} + \sqrt{5})}{(\sqrt{2} - \sqrt{5})(\sqrt{2} + \sqrt{5})}$.
Step 3:
Use the difference of squares formula to expand the denominator: $\frac{(\sqrt{2} + \sqrt{5})^2}{(\sqrt{2})^2 - (\sqrt{5})^2}$.
Step 4:
Simplify the denominator: $\frac{(\sqrt{2} + \sqrt{5})^2}{-3}$.
Step 5:
Simplify the numerator by raising the binomial to the second power.
Step 5.1:
Raise the binomial to the power of 1: $\frac{(\sqrt{2} + \sqrt{5})^1(\sqrt{2} + \sqrt{5})}{-3}$.
Step 5.2:
Raise the binomial to the power of 1 again: $\frac{(\sqrt{2} + \sqrt{5})^1(\sqrt{2} + \sqrt{5})^1}{-3}$.
Step 5.3:
Combine the exponents using the power rule: $\frac{(\sqrt{2} + \sqrt{5})^{1+1}}{-3}$.
Step 5.4:
Add the exponents: $\frac{(\sqrt{2} + \sqrt{5})^2}{-3}$.
Step 6:
Rewrite the squared binomial: $\frac{(\sqrt{2} + \sqrt{5})(\sqrt{2} + \sqrt{5})}{-3}$.
Step 7:
Expand the binomial using the FOIL method.
Step 7.1:
Distribute the first term: $\frac{\sqrt{2}(\sqrt{2} + \sqrt{5}) + \sqrt{5}(\sqrt{2} + \sqrt{5})}{-3}$.
Step 7.2:
Distribute the second term: $\frac{\sqrt{2}\sqrt{2} + \sqrt{2}\sqrt{5} + \sqrt{5}(\sqrt{2} + \sqrt{5})}{-3}$.
Step 7.3:
Distribute the remaining terms: $\frac{\sqrt{2}\sqrt{2} + \sqrt{2}\sqrt{5} + \sqrt{5}\sqrt{2} + \sqrt{5}\sqrt{5}}{-3}$.
Step 8:
Combine like terms and simplify.
Step 8.1:
Simplify each term.
Step 8.1.1:
Combine using the product rule for radicals: $\frac{\sqrt{2 \cdot 2} + \sqrt{2}\sqrt{5} + \sqrt{5}\sqrt{2} + \sqrt{5}\sqrt{5}}{-3}$.
Step 8.1.2:
Multiply the radicals: $\frac{\sqrt{4} + \sqrt{2}\sqrt{5} + \sqrt{5}\sqrt{2} + \sqrt{5}\sqrt{5}}{-3}$.
Step 8.1.3:
Rewrite 4 as $2^2$: $\frac{\sqrt{2^2} + \sqrt{2}\sqrt{5} + \sqrt{5}\sqrt{2} + \sqrt{5}\sqrt{5}}{-3}$.
Step 8.1.4:
Extract square roots: $\frac{2 + \sqrt{2}\sqrt{5} + \sqrt{5}\sqrt{2} + \sqrt{5}\sqrt{5}}{-3}$.
Step 8.1.5:
Combine using the product rule for radicals: $\frac{2 + \sqrt{2 \cdot 5} + \sqrt{5}\sqrt{2} + \sqrt{5}\sqrt{5}}{-3}$.
Step 8.1.6:
Multiply the radicals: $\frac{2 + \sqrt{10} + \sqrt{5}\sqrt{2} + \sqrt{5}\sqrt{5}}{-3}$.
Step 8.1.7:
Combine using the product rule for radicals: $\frac{2 + \sqrt{10} + \sqrt{5 \cdot 2} + \sqrt{5}\sqrt{5}}{-3}$.
Step 8.1.8:
Multiply the radicals: $\frac{2 + \sqrt{10} + \sqrt{10} + \sqrt{5}\sqrt{5}}{-3}$.
Step 8.1.9:
Combine using the product rule for radicals: $\frac{2 + \sqrt{10} + \sqrt{10} + \sqrt{5 \cdot 5}}{-3}$.
Step 8.1.10:
Multiply the radicals: $\frac{2 + \sqrt{10} + \sqrt{10} + \sqrt{25}}{-3}$.
Step 8.1.11:
Rewrite 25 as $5^2$: $\frac{2 + \sqrt{10} + \sqrt{10} + \sqrt{5^2}}{-3}$.
Step 8.1.12:
Extract square roots: $\frac{2 + \sqrt{10} + \sqrt{10} + 5}{-3}$.
Step 8.2:
Add the integers: $\frac{7 + \sqrt{10} + \sqrt{10}}{-3}$.
Step 8.3:
Combine the square roots: $\frac{7 + 2\sqrt{10}}{-3}$.
Step 9:
Place the negative sign in front of the fraction: $-\frac{7 + 2\sqrt{10}}{3}$.
Knowledge Notes:
The problem involves simplifying a complex fraction that contains square roots. To do this, several mathematical concepts and rules are used:
Conjugate: The conjugate of a binomial $a + b$ is $a - b$. Multiplying a binomial by its conjugate results in the difference of squares, which is a technique used to rationalize denominators containing square roots.
Difference of Squares: This is a pattern where $(a + b)(a - b) = a^2 - b^2$. It is used to simplify expressions where the denominator is a binomial with square roots.
FOIL Method: This stands for First, Outer, Inner, Last and is a technique for multiplying two binomials. It ensures that each term in the first binomial is multiplied by each term in the second binomial.
Simplifying Square Roots: When a number under a square root is a perfect square, it can be simplified by taking the square root of that number.
Combining Like Terms: Terms that are similar (e.g., have the same variable and exponent) can be combined by adding or subtracting their coefficients.
Product Rule for Radicals: This rule states that $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$. It allows the multiplication of square roots to be simplified.
Negative Fractions: A negative sign in a fraction can be placed in front of the numerator, in front of the denominator, or in front of the entire fraction without changing the value of the fraction.
