Simplify ( square root of 3/13)/( square root of 18)

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: $0.11322770\dots$

Knowledge Notes:

To solve the given problem, we used several mathematical concepts and rules:

1. Radical Simplification: The square root of a fraction can be expressed as the fraction of the square roots of the numerator and the denominator.

2. Rationalizing the Denominator: This involves multiplying the numerator and the denominator by a conjugate to eliminate the square root from the denominator.

3. 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.

4. 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.

5. Factoring: Breaking down a number into its prime factors can help simplify square roots.

6. Multiplying Fractions: To multiply fractions, we multiply the numerators together and the denominators together.

7. 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.