Simplify ( square root of 5-2 square root of 2)/( square root of 2-2 square root of 5)
The given problem is an algebraic expression that requires simplification. It involves manipulating a fraction with radical expressions (square roots) in both the numerator and the denominator. The task is to perform operations on the expression to rewrite it in a simpler or more conventional form, typically by rationalizing the denominator. This means that the square roots in the denominator should be eliminated if possible, to leave an expression with a rational number in the denominator. The process will likely involve the multiplication of both the numerator and the denominator by a suitable conjugate to clear the radicals.
$\frac{\sqrt{5} - 2 \sqrt{2}}{\sqrt{2} - 2 \sqrt{5}}$
Answer
Solution:
Step:1
Rationalize the denominator of $\frac{\sqrt{5} - 2\sqrt{2}}{\sqrt{2} - 2\sqrt{5}}$ by multiplying by the conjugate $\frac{\sqrt{2} + 2\sqrt{5}}{\sqrt{2} + 2\sqrt{5}}$.
Step:2
Combine the fractions into one.
Step:2.1
Multiply the numerators and denominators: $\frac{(\sqrt{5} - 2\sqrt{2})(\sqrt{2} + 2\sqrt{5})}{(\sqrt{2} - 2\sqrt{5})(\sqrt{2} + 2\sqrt{5})}$.
Step:2.2
Use the difference of squares to expand the denominator: $\frac{(\sqrt{5} - 2\sqrt{2})(\sqrt{2} + 2\sqrt{5})}{(\sqrt{2})^2 - (2\sqrt{5})^2}$.
Step:2.3
Simplify the denominator: $\frac{(\sqrt{5} - 2\sqrt{2})(\sqrt{2} + 2\sqrt{5})}{2 - 20} = \frac{(\sqrt{5} - 2\sqrt{2})(\sqrt{2} + 2\sqrt{5})}{-18}$.
Step:3
Expand the numerator using the distributive property (FOIL).
Step:3.1
Distribute $\sqrt{5}$ and $-2\sqrt{2}$ across $\sqrt{2} + 2\sqrt{5}$.
Step:3.2
Continue distributing: $\sqrt{5}\sqrt{2} + 2\sqrt{5}^2 - 2\sqrt{2}\sqrt{2} - 4\sqrt{2}\sqrt{5}$.
Step:3.3
Complete the distribution: $\sqrt{10} + 2(5) - 2(2) - 4\sqrt{10}$.
Step:4
Combine like terms and simplify.
Step:4.1
Simplify each term: $\frac{\sqrt{10} + 10 - 4 - 4\sqrt{10}}{-18}$.
Step:4.2
Combine the radical terms: $\frac{10 - 4 - 3\sqrt{10}}{-18}$.
Step:4.3
Combine the integer terms: $\frac{6 - 3\sqrt{10}}{-18}$.
Step:5
Reduce the fraction by canceling common factors.
Step:5.1
Factor out a $3$ from the numerator: $\frac{3(2 - \sqrt{10})}{-18}$.
Step:5.2
Factor out a $3$ from the denominator: $\frac{3(2 - \sqrt{10})}{3(-6)}$.
Step:5.3
Cancel the common factor of $3$: $\frac{2 - \sqrt{10}}{-6}$.
Step:6
Move the negative sign in front of the fraction: $-\frac{2 - \sqrt{10}}{6}$.
Step:7
The final result can be expressed in different forms:
- Exact Form: $-\frac{2 - \sqrt{10}}{6}$
- Decimal Form: Approximately $0.19371294$
Knowledge Notes:
Rationalizing the Denominator: This technique involves multiplying the numerator and denominator by a conjugate to eliminate radicals from the denominator.
Conjugate: The conjugate of a binomial $a + b$ is $a - b$. When a binomial is multiplied by its conjugate, the result is a difference of squares.
Difference of Squares: This is a pattern used in algebra where $(a + b)(a - b) = a^2 - b^2$.
FOIL Method: An acronym for First, Outer, Inner, Last, which is a technique used to multiply two binomials.
Distributive Property: This property states that $a(b + c) = ab + ac$. It is used to expand expressions.
Simplifying Radicals: Combining like terms that involve square roots and simplifying expressions with square roots.
Reducing Fractions: This involves dividing the numerator and the denominator by their greatest common factor to simplify the fraction.
Exact vs. Decimal Form: Exact form maintains the radical and fraction form of the answer, while decimal form provides an approximate value.
