Simplify ( square root of 3/13)/( square root of 18)
The given problem is a mathematical expression that requires simplification. Specifically, it asks for the simplification of a fraction where both the numerator and the denominator contain square roots. The numerator is the square root of the fraction 3/13, and the denominator is the square root of the integer 18. The task is to perform the necessary algebraic manipulations to simplify this expression to its most reduced form, which might involve rationalizing the denominator and simplifying any square roots or fractions where possible.
$\frac{\sqrt{\frac{3}{13}}}{\sqrt{18}}$
Answer
Solution:
Step:1
Simplify the numerator.
Step:1.1
Express $\sqrt{\frac{3}{13}}$ as $\frac{\sqrt{3}}{\sqrt{13}}$.$\frac{\frac{\sqrt{3}}{\sqrt{13}}}{\sqrt{18}}$
Step:1.2
Multiply $\frac{\sqrt{3}}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$.$\frac{\frac{\sqrt{3}}{\sqrt{13}} \cdot \frac{\sqrt{13}}{\sqrt{13}}}{\sqrt{18}}$
Step:1.3
Simplify the fraction in the numerator.
Step:1.3.1
Multiply $\frac{\sqrt{3}}{\sqrt{13}}$ by $\frac{\sqrt{13}}{\sqrt{13}}$.$\frac{\frac{\sqrt{3} \cdot \sqrt{13}}{\sqrt{13} \cdot \sqrt{13}}}{\sqrt{18}}$
Step:1.3.2
Simplify the square root of $13$.$\frac{\frac{\sqrt{3} \cdot \sqrt{13}}{(\sqrt{13})^2}}{\sqrt{18}}$
Step:1.3.3
Simplify the denominator using the exponent rule.$\frac{\frac{\sqrt{3} \cdot \sqrt{13}}{13}}{\sqrt{18}}$
Step:1.4
Combine the square roots in the numerator.
Step:1.4.1
Use the product rule for radicals.$\frac{\sqrt{3 \cdot 13}}{13}{\sqrt{18}}$
Step:1.4.2
Calculate the product under the radical.$\frac{\sqrt{39}}{13}{\sqrt{18}}$
Step:2
Simplify the denominator.
Step:2.1
Factor the number under the square root.
Step:2.1.1
Factor $9$ from $18$.$\frac{\sqrt{39}}{13}{\sqrt{9 \cdot 2}}$
Step:2.1.2
Express $9$ as $3^2$.$\frac{\sqrt{39}}{13}{\sqrt{3^2 \cdot 2}}$
Step:2.2
Extract square roots from the radical.
Step:2.2.1
Extract $3$ from under the radical.$\frac{\sqrt{39}}{13}{3\sqrt{2}}$
Step:3
Multiply the numerator by the reciprocal of the denominator.$\frac{\sqrt{39}}{13} \cdot \frac{1}{3\sqrt{2}}$
Step:4
Rationalize the denominator.
Step:4.1
Multiply by $\frac{\sqrt{2}}{\sqrt{2}}$.$\frac{\sqrt{39}}{13} \cdot \frac{\sqrt{2}}{3\sqrt{2} \cdot \sqrt{2}}$
Step:4.2
Simplify the denominator.
Step:4.2.1
Simplify using the power of a power rule.$\frac{\sqrt{39}}{13} \cdot \frac{\sqrt{2}}{3 \cdot 2}$
Step:4.2.2
Calculate the product in the denominator.$\frac{\sqrt{39}}{13} \cdot \frac{\sqrt{2}}{6}$
Step:5
Multiply the fractions.
Step:5.1
Multiply the numerators.$\frac{\sqrt{39} \cdot \sqrt{2}}{13 \cdot 6}$
Step:5.2
Apply the product rule for radicals.$\frac{\sqrt{39 \cdot 2}}{13 \cdot 6}$
Step:5.3
Calculate the product under the radical and in the denominator.$\frac{\sqrt{78}}{78}$
Step:6
Write the final result in different forms.
Exact Form: $\frac{\sqrt{78}}{78}$ Decimal Form: $0.11322770 \ldots$
Knowledge Notes:
To solve the given problem, we used several mathematical concepts and rules:
Radical Simplification: The square root of a fraction can be expressed as the fraction of the square roots of the numerator and the denominator.
Rationalizing the Denominator: This involves multiplying the numerator and the denominator by a conjugate to eliminate the square root from the denominator.
Product Rule for Radicals: $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$. This rule allows us to combine or separate products under a single radical.
Exponent Rules: These include the power of a power rule and the power of a product rule, which are used to simplify expressions involving exponents.
Factoring: Breaking down a number into its prime factors can help simplify square roots.
Multiplying Fractions: To multiply fractions, we multiply the numerators together and the denominators together.
Decimal Representation: The exact form of a radical can be approximated by a decimal, which is useful for practical applications.
Throughout the solution, we applied these rules to simplify the given expression step by step.
