Simplify ( square root of 7b)/( square root of 18)

Solution:

Step 1: Simplify the denominator.

• Step 1.1: Express 18 as the product of its prime factors.

  • Step 1.1.1: Decompose 18 into 9 and 2. $\frac{\sqrt{7b}}{\sqrt{9 \cdot 2}}$
  • Step 1.1.2: Replace 9 with its square form, $3^2$. $\frac{\sqrt{7b}}{\sqrt{3^2 \cdot 2}}$

• Step 1.2: Extract the square root of the perfect square from the radical. $\frac{\sqrt{7b}}{3\sqrt{2}}$

Step 2: Rationalize the denominator.

• Multiply the fraction by $\frac{\sqrt{2}}{\sqrt{2}}$. $\frac{\sqrt{7b}}{3\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 3: Simplify the denominator further.

• Step 3.1: Multiply the numerators and denominators. $\frac{\sqrt{7b}\sqrt{2}}{3\sqrt{2}\sqrt{2}}$

• Step 3.2: Combine the square roots in the denominator. $\frac{\sqrt{7b}\sqrt{2}}{3(\sqrt{2}\sqrt{2})}$

• Step 3.3: Recognize that $\sqrt{2}$ raised to the power of 1 is still $\sqrt{2}$. $\frac{\sqrt{7b}\sqrt{2}}{3((\sqrt{2})^1\sqrt{2})}$

• Step 3.4: Apply the same recognition to the second $\sqrt{2}$. $\frac{\sqrt{7b}\sqrt{2}}{3((\sqrt{2})^1(\sqrt{2})^1)}$

• Step 3.5: Use the exponent multiplication rule to combine the roots. $\frac{\sqrt{7b}\sqrt{2}}{3(\sqrt{2})^{1+1}}$

• Step 3.6: Add the exponents. $\frac{\sqrt{7b}\sqrt{2}}{3(\sqrt{2})^2}$

• Step 3.7: Simplify the square of $\sqrt{2}$ to 2.

  • Step 3.7.1: Express $\sqrt{2}$ as $2^{\frac{1}{2}}$. $\frac{\sqrt{7b}\sqrt{2}}{3((2^{\frac{1}{2}})^2)}$

  • Step 3.7.2: Apply the exponent rule to the power of a power. $\frac{\sqrt{7b}\sqrt{2}}{3 \cdot 2^{\frac{1}{2} \cdot 2}}$

  • Step 3.7.3: Multiply the exponents. $\frac{\sqrt{7b}\sqrt{2}}{3 \cdot 2^{\frac{2}{2}}}$

  • Step 3.7.4: Simplify the expression by canceling the common factors.

    • Step 3.7.4.1: Cancel out the common factors. $\frac{\sqrt{7b}\sqrt{2}}{3 \cdot 2^{\cancel{2}/\cancel{2}}}$
    • Step 3.7.4.2: Rewrite the simplified expression. $\frac{\sqrt{7b}\sqrt{2}}{3 \cdot 2^1}$

  • Step 3.7.5: Evaluate the exponent. $\frac{\sqrt{7b}\sqrt{2}}{3 \cdot 2}$

Step 4: Simplify the numerator.

• Step 4.1: Apply the product rule for radicals. $\frac{\sqrt{7b \cdot 2}}{3 \cdot 2}$
• Step 4.2: Perform the multiplication inside the radical. $\frac{\sqrt{14b}}{3 \cdot 2}$

Step 5: Final simplification.

• Multiply the numbers outside the radical. $\frac{\sqrt{14b}}{6}$

Knowledge Notes:

To simplify a fraction involving square roots, we can use the following steps and rules:

1. Prime Factorization: Decompose numbers into their prime factors to identify and extract perfect squares.

2. Square Root Extraction: For any perfect square under a square root, we can take the square root and remove it from under the radical.

3. Rationalizing the Denominator: If the denominator contains a square root, we multiply the fraction by a form of 1 that will eliminate the square root from the denominator.

4. Simplification of Radicals: Combine radicals using the product rule, $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$, and simplify the expression.

5. Exponent Rules: Apply exponent rules such as $(a^m)^n = a^{mn}$ and $a^{m} a^{n} = a^{m + n}$ to simplify expressions with powers.

6. Simplification: Cancel common factors and simplify the expression to its lowest terms.

In this problem, we applied these steps to simplify the given expression by rationalizing the denominator and simplifying the radicals.