Simplify ( square root of 40-2 square root of 5)/( square root of 10)
The question is asking to simplify an expression involving square roots. Specifically, it gives you a fraction where the numerator is composed of the square root of 40 minus twice the square root of 5, and the denominator is the square root of 10. The task is to perform the simplification using algebraic rules pertinent to square roots and fractions, to express the result in its simplest form.
$\frac{\sqrt{40} - 2 \sqrt{5}}{\sqrt{10}}$
Answer
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 $1.58578643 \ldots$
Knowledge Notes:
To simplify the given expression, we used several algebraic techniques and properties of radicals:
Factorization: Breaking down numbers into their prime factors to simplify expressions under a radical.
Rationalizing the Denominator: Multiplying the numerator and denominator by a suitable expression to remove the radical from the denominator.
Power Rule: Using the property $a^{m} a^{n} = a^{m + n}$ to combine and simplify expressions with the same base.
Product Rule for Radicals: $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$ allows us to combine or separate radicals.
Distributive Property: This property is used to multiply a single term and two or more terms inside a set of parentheses.
Reducing Common Factors: Dividing the numerator and denominator by common factors to simplify fractions.
Square Roots of Perfect Squares: Recognizing and simplifying the square roots of perfect squares, such as $\sqrt{100} = 10$.
Decimal Approximation: Converting an exact form into a decimal approximation for practical use.
