Simplify ( square root of 40-2 square root of 5)/( square root of 10)

Solution:

Step 1: Simplify the numerator.

• Step 1.1: Express $40$ as the product of $2^2$ and $10$.

  • Step 1.1.1: Extract the square root of $4$ from $40$. $\frac{\sqrt{4\cdot 10}-2\sqrt{5}}{\sqrt{10}}$
  • Step 1.1.2: Represent $4$ as $2^2$. $\frac{\sqrt{2^2\cdot 10}-2\sqrt{5}}{\sqrt{10}}$

• Step 1.2: Remove terms from under the radical sign. $\frac{2\sqrt{10}-2\sqrt{5}}{\sqrt{10}}$

Step 2: Rationalize the denominator.

• Multiply the expression by $\frac{\sqrt{10}}{\sqrt{10}}$. $\frac{2\sqrt{10}-2\sqrt{5}}{\sqrt{10}}\cdot \frac{\sqrt{10}}{\sqrt{10}}$

Step 3: Simplify the denominator.

• Step 3.1: Multiply the numerator by $\sqrt{10}$. $\frac{(2\sqrt{10}-2\sqrt{5})\sqrt{10}}{\sqrt{10}\sqrt{10}}$

• Step 3.2 - 3.5: Apply the power rule to combine the square roots in the denominator. $\frac{(2\sqrt{10}-2\sqrt{5})\sqrt{10}}{(\sqrt{10}{)}^2}$

• Step 3.6: Convert $(\sqrt{10}{)}^2$ to $10$.

  • Step 3.6.1 - 3.6.4: Simplify the power expression. $\frac{(2\sqrt{10}-2\sqrt{5})\sqrt{10}}{10}$

Step 4: Reduce common factors.

• Step 4.1: Factor out a $2$ from the numerator. $\frac{2(\sqrt{10}-\sqrt{5})\sqrt{10}}{10}$
• Step 4.2: Simplify by canceling out the common factor of $2$. $\frac{(\sqrt{10}-\sqrt{5})\sqrt{10}}{5}$

Step 5: Distribute the square root of $10$.

• $\frac{\sqrt{10}\cdot \sqrt{10}-\sqrt{5}\cdot \sqrt{10}}{5}$

Step 6: Apply the product rule for radicals.

• $\frac{\sqrt{10\cdot 10}-\sqrt{5\cdot 10}}{5}$

Step 7: Simplify the radical expressions.

• Step 7.1: Combine radicals using the product rule. $\frac{\sqrt{10\cdot 10}-\sqrt{50}}{5}$
• Step 7.2: Calculate the product inside the radical. $\frac{\sqrt{100}-\sqrt{50}}{5}$

Step 8: Simplify each term.

• Step 8.1 - 8.3: Extract the square root of $100$. $\frac{10-\sqrt{50}}{5}$

• Step 8.4: Express $50$ as $5^2\cdot 2$.

  • Step 8.4.1 - 8.5: Pull the square root of $25$ out of the radical. $\frac{10-5\sqrt{2}}{5}$

Step 9: Cancel common factors.

• Step 9.1 - 9.3: Factor out a $5$ from the numerator. $\frac{5(2-\sqrt{2})}{5}$
• Step 9.4: Simplify by canceling out the common factor of $5$. $\frac{2-\sqrt{2}}{1}$

Step 10: Present the result in various forms.

• Exact Form: $2-\sqrt{2}$
• Decimal Form: Approximately $1.58578643\dots$

Knowledge Notes:

To simplify the given expression, we used several algebraic techniques and properties of radicals:

1. Factorization: Breaking down numbers into their prime factors to simplify expressions under a radical.

2. Rationalizing the Denominator: Multiplying the numerator and denominator by a suitable expression to remove the radical from the denominator.

3. Power Rule: Using the property $a^m{a}^n=a^{m+n}$ to combine and simplify expressions with the same base.

4. Product Rule for Radicals: $\sqrt{a}\cdot \sqrt{b}=\sqrt{a\cdot b}$ allows us to combine or separate radicals.

5. Distributive Property: This property is used to multiply a single term and two or more terms inside a set of parentheses.

6. Reducing Common Factors: Dividing the numerator and denominator by common factors to simplify fractions.

7. Square Roots of Perfect Squares: Recognizing and simplifying the square roots of perfect squares, such as $\sqrt{100}=10$.

8. Decimal Approximation: Converting an exact form into a decimal approximation for practical use.