Simplify ( square root of n)/( square root of 112)

Solution:

Step 1: Simplify the denominator.

• Step 1.1: Express $112$ as a product of its prime factors.

  • Step 1.1.1: Extract $16$ from $112$, resulting in $\frac{\sqrt{n}}{\sqrt{16\cdot 7}}$.

  • Step 1.1.2: Recognize that $16$ is $4^2$, thus $\frac{\sqrt{n}}{\sqrt{4^2\cdot 7}}$.

• Step 1.2: Remove the perfect square from under the radical, giving $\frac{\sqrt{n}}{4\sqrt{7}}$.

Step 2: Rationalize the denominator.

• Multiply the fraction by $\frac{\sqrt{7}}{\sqrt{7}}$ to get $\frac{\sqrt{n}}{4\sqrt{7}}\cdot \frac{\sqrt{7}}{\sqrt{7}}$.

Step 3: Simplify the denominator further.

• Step 3.1: Multiply the numerators and denominators to get $\frac{\sqrt{n}\sqrt{7}}{4\sqrt{7}\sqrt{7}}$.

• Step 3.2: Combine the square roots in the denominator.

• Step 3.3: Recognize that raising a square root to the power of $1$ does not change its value.

• Step 3.4: Apply the same principle to the other square root.

• Step 3.5: Use the exponent rule $a^m{a}^n=a^{m+n}$ to combine the square roots into $\frac{\sqrt{n}\sqrt{7}}{4(\sqrt{7}{)}^{1+1}}$.

• Step 3.6: Add the exponents to get $\frac{\sqrt{n}\sqrt{7}}{4(\sqrt{7}{)}^2}$.

• Step 3.7: Convert the squared square root back to its base number.

  • Step 3.7.1: Rewrite $\sqrt{7}$ as $7^{\frac{1}{2}}$.

  • Step 3.7.2: Apply the power rule $(a^m{)}^n=a^{mn}$ to get $\frac{\sqrt{n}\sqrt{7}}{4\cdot 7^{\frac{1}{2}\cdot 2}}$.

  • Step 3.7.3: Simplify the exponent to get $\frac{\sqrt{n}\sqrt{7}}{4\cdot 7^{\frac{2}{2}}}$.

  • Step 3.7.4: Reduce the fraction in the exponent.

    • Step 3.7.4.1: Simplify to get $\frac{\sqrt{n}\sqrt{7}}{4\cdot 7^{\frac{\cancel{2}}{\cancel{2}}}}$.

    • Step 3.7.4.2: Rewrite the expression as $\frac{\sqrt{n}\sqrt{7}}{4\cdot 7^1}$.

  • Step 3.7.5: Evaluate the exponent to get $\frac{\sqrt{n}\sqrt{7}}{4\cdot 7}$.

Step 4: Combine the radicals.

• Use the product rule for radicals to combine them into $\frac{\sqrt{n\cdot 7}}{4\cdot 7}$.

Step 5: Final simplification.

• Step 5.1: Multiply the numbers in the denominator to get $\frac{\sqrt{n\cdot 7}}{28}$.

• Step 5.2: Rearrange the factors under the radical if necessary to get $\frac{\sqrt{7n}}{28}$.

Knowledge Notes:

To solve the given problem, we use several mathematical properties and rules:

1. Prime Factorization: Breaking down a number into its prime factors helps to simplify the square root of that number.

2. Square Root Simplification: The square root of a product of a perfect square and another number can be simplified by taking the square root of the perfect square outside the radical.

3. Rationalizing the Denominator: Multiplying the numerator and denominator by the square root that is present in the denominator eliminates the radical from the denominator.

4. Product Rule for Radicals: $\sqrt{a}\cdot \sqrt{b}=\sqrt{ab}$ allows us to combine square roots.

5. Power Rule for Exponents: $a^m\cdot a^n=a^{m+n}$ and $(a^m{)}^n=a^{mn}$ are used to simplify expressions with exponents.

6. Simplifying Exponents: When an exponent is a fraction, such as $\frac{2}{2}$, it can be simplified to $1$, and $a^1=a$.

7. Combining Radicals and Simplification: After rationalizing and simplifying, we combine the radicals and simplify the expression to its lowest terms.

These rules and properties are fundamental in algebra and are used extensively in simplifying expressions involving square roots and exponents.