Simplify ( square root of 5-2 square root of 2)/( square root of 2-2 square root of 5)

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:

1. Rationalizing the Denominator: This technique involves multiplying the numerator and denominator by a conjugate to eliminate radicals from the denominator.

2. 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.

3. Difference of Squares: This is a pattern used in algebra where $(a+b)(a-b)=a^2-b^2$.

4. FOIL Method: An acronym for First, Outer, Inner, Last, which is a technique used to multiply two binomials.

5. Distributive Property: This property states that $a(b+c)=ab+ac$. It is used to expand expressions.

6. Simplifying Radicals: Combining like terms that involve square roots and simplifying expressions with square roots.

7. Reducing Fractions: This involves dividing the numerator and the denominator by their greatest common factor to simplify the fraction.

8. Exact vs. Decimal Form: Exact form maintains the radical and fraction form of the answer, while decimal form provides an approximate value.