Simplify (3+ square root of 18)/(1+ square root of 8)
Your question asks for the simplification of a mathematical expression that contains radicals (square roots). Specifically, it involves the simplification of a rational expression where the numerator is the sum of a number and the square root of 18, and the denominator is the sum of 1 and the square root of 8. The objective is to manipulate this expression to get to a simpler form, potentially by rationalizing the denominator, which means getting rid of the square root in the denominator by multiplying both the numerator and the denominator by a suitable expression.
$\frac{3 + \sqrt{18}}{1 + \sqrt{8}}$
Answer
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:
Simplifying Square Roots: Expressing numbers under the square root as a product of squares and other numbers to simplify the square root.
Rationalizing the Denominator: Multiplying the numerator and the denominator by the conjugate of the denominator to eliminate the square root from the denominator.
FOIL Method: Stands for First, Outer, Inner, Last, which is a technique used to expand the product of two binomials.
Difference of Squares: Recognizing that $(a + b)(a - b) = a^2 - b^2$ and applying it to simplify expressions.
Distributive Property: Applying the property $a(b + c) = ab + ac$ to expand expressions.
Combining Like Terms: Grouping and simplifying terms that have the same variable to the same power.
Power Rules: Using rules like $a^m a^n = a^{m+n}$ and $(a^m)^n = a^{mn}$ to simplify expressions with exponents.
Simplification: The process of reducing an expression to its simplest form by performing all possible operations and combining like terms.
