Simplify (3+ square root of 18)/(1+ square root of 8)

Solution:

Step:1

Refine the numerator.

Step:1.1

Express $18$ as $9\times 2$.

Step:1.1.1

Extract $9$ from $18$. $\frac{3+\sqrt{9\cdot 2}}{1+\sqrt{8}}$

Step:1.1.2

Represent $9$ as $(3^2)$. $\frac{3+\sqrt{3^2\cdot 2}}{1+\sqrt{8}}$

Step:1.2

Extract square roots where possible. $\frac{3+3\sqrt{2}}{1+\sqrt{8}}$

Step:2

Streamline the denominator.

Step:2.1

Express $8$ as $4\times 2$.

Step:2.1.1

Extract $4$ from $8$. $\frac{3+3\sqrt{2}}{1+\sqrt{4\cdot 2}}$

Step:2.1.2

Represent $4$ as $(2^2)$. $\frac{3+3\sqrt{2}}{1+\sqrt{2^2\cdot 2}}$

Step:2.2

Extract square roots where possible. $\frac{3+3\sqrt{2}}{1+2\sqrt{2}}$

Step:3

Multiply $\frac{3+3\sqrt{2}}{1+2\sqrt{2}}$ by the conjugate $\frac{1-2\sqrt{2}}{1-2\sqrt{2}}$.

Step:4

Consolidate the fractions.

Step:4.1

Multiply the numerators. $\frac{(3+3\sqrt{2})(1-2\sqrt{2})}{(1+2\sqrt{2})(1-2\sqrt{2})}$

Step:4.2

Expand the denominator using the difference of squares. $\frac{(3+3\sqrt{2})(1-2\sqrt{2})}{1-(2\sqrt{2}{)}^2}$

Step:4.3

Simplify the denominator. $\frac{(3+3\sqrt{2})(1-2\sqrt{2})}{-7}$

Step:5

Expand the numerator using the distributive property.

Step:5.1

Distribute terms. $\frac{3(1-2\sqrt{2})+3\sqrt{2}(1-2\sqrt{2})}{-7}$

Step:5.2

Further distribute. $\frac{3-6\sqrt{2}+3\sqrt{2}(1-2\sqrt{2})}{-7}$

Step:5.3

Complete the distribution. $\frac{3-6\sqrt{2}+3\sqrt{2}-6(2)}{-7}$

Step:6

Combine like terms and simplify.

Step:6.1

Simplify each term.

Step:6.1.1

Multiply $3$ by $1$. $\frac{3-6\sqrt{2}+3\sqrt{2}-12}{-7}$

Step:6.2

Combine numerical terms. $\frac{-9-6\sqrt{2}+3\sqrt{2}}{-7}$

Step:6.3

Combine like radical terms. $\frac{-9-3\sqrt{2}}{-7}$

Step:7

Remove the negative from the fraction. $\frac{9+3\sqrt{2}}{7}$

Step:8

The expression is already simplified.

Step:9

No further factoring is necessary.

Step:10

The expression remains unchanged.

Step:11

Final simplification is not needed.

Step:12

The result in various forms:

Exact Form: $\frac{9+3\sqrt{2}}{7}$ Decimal Form: Approximately $1.8918$

Knowledge Notes:

The problem-solving process involves simplifying a complex fraction by rationalizing the denominator. This is done by multiplying the fraction by the conjugate of the denominator to eliminate the square root from the denominator. The steps include:

1. Simplifying Square Roots: Expressing numbers under the square root as a product of squares and other numbers to simplify the square root.

2. Rationalizing the Denominator: Multiplying the numerator and the denominator by the conjugate of the denominator to eliminate the square root from the denominator.

3. FOIL Method: Stands for First, Outer, Inner, Last, which is a technique used to expand the product of two binomials.

4. Difference of Squares: Recognizing that $(a+b)(a-b)=a^2-b^2$ and applying it to simplify expressions.

5. Distributive Property: Applying the property $a(b+c)=ab+ac$ to expand expressions.

6. Combining Like Terms: Grouping and simplifying terms that have the same variable to the same power.

7. Power Rules: Using rules like $a^m{a}^n=a^{m+n}$ and $(a^m{)}^n=a^{mn}$ to simplify expressions with exponents.

8. Simplification: The process of reducing an expression to its simplest form by performing all possible operations and combining like terms.