Simplify ( square root of 1/2)( square root of 3/5)

Solution:

Step 1:

Apply the product rule for radicals to combine the square roots: $\sqrt{\frac{1}{2} \cdot \frac{3}{5}}$

Step 2:

Perform the multiplication of the fractions: $\sqrt{\frac{1 \cdot 3}{2 \cdot 5}}$

Step 3:

Calculate the product of the denominators: $\sqrt{\frac{3}{10}}$

Step 4:

Separate the radical over the numerator and denominator: $\frac{\sqrt{3}}{\sqrt{10}}$

Step 5:

Rationalize the denominator by multiplying by $\frac{\sqrt{10}}{\sqrt{10}}$: $\frac{\sqrt{3}}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}}$

Step 6:

Simplify the expression under the radical in the denominator.

Step 6.1:

Multiply the radicals in the numerator and denominator: $\frac{\sqrt{3} \cdot \sqrt{10}}{\sqrt{10} \cdot \sqrt{10}}$

Step 6.2:

Express $\sqrt{10}$ as a power: $\frac{\sqrt{3} \cdot \sqrt{10}}{(\sqrt{10})^{1} \cdot \sqrt{10}}$

Step 6.3:

Rewrite the denominator using the same base: $\frac{\sqrt{3} \cdot \sqrt{10}}{(\sqrt{10})^{1} \cdot (\sqrt{10})^{1}}$

Step 6.4:

Combine the exponents in the denominator using the power rule: $\frac{\sqrt{3} \cdot \sqrt{10}}{(\sqrt{10})^{1 + 1}}$

Step 6.5:

Add the exponents in the denominator: $\frac{\sqrt{3} \cdot \sqrt{10}}{(\sqrt{10})^{2}}$

Step 6.6:

Simplify the denominator to its base form.

Step 6.6.1:

Express $\sqrt{10}$ as a power of 10: $\frac{\sqrt{3} \cdot \sqrt{10}}{((10)^{\frac{1}{2}})^{2}}$

Step 6.6.2:

Apply the power rule to the denominator: $\frac{\sqrt{3} \cdot \sqrt{10}}{(10)^{\frac{1}{2} \cdot 2}}$

Step 6.6.3:

Simplify the exponent in the denominator: $\frac{\sqrt{3} \cdot \sqrt{10}}{(10)^{\frac{2}{2}}}$

Step 6.6.4:

Cancel the common factors in the exponent.

Step 6.6.4.1:

Cancel out the common factor: $\frac{\sqrt{3} \cdot \sqrt{10}}{(10)^{\cancel{2}/\cancel{2}}}$

Step 6.6.4.2:

Rewrite the denominator as a whole number: $\frac{\sqrt{3} \cdot \sqrt{10}}{10}$

Step 6.6.5:

Evaluate the simplified denominator: $\frac{\sqrt{3} \cdot \sqrt{10}}{10}$

Step 7:

Simplify the numerator by combining the radicals.

Step 7.1:

Use the product rule for radicals: $\frac{\sqrt{3 \cdot 10}}{10}$

Step 7.2:

Calculate the product under the radical: $\frac{\sqrt{30}}{10}$

Step 8:

Present the result in various forms.

Exact Form: $\frac{\sqrt{30}}{10}$

Decimal Form: $0.5477225575051661 \ldots$

Knowledge Notes:

The problem-solving process involves simplifying the product of two square roots. Here are the relevant knowledge points and detailed explanations:

1. Product Rule for Radicals: The product rule states that $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$. This allows us to combine or separate square roots when multiplying or dividing.

2. Multiplying Fractions: To multiply fractions, multiply the numerators together and the denominators together: $\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$.

3. Rationalizing the Denominator: When a radical is in the denominator, we can make it rational (a non-radical) by multiplying the numerator and denominator by a suitable radical. This process is called rationalizing the denominator.

4. Power Rule: The power rule for exponents states that $a^m \cdot a^n = a^{m+n}$. This is used to combine like bases with exponents.

5. Simplifying Radicals: When simplifying radicals, we look for factors that are perfect squares and simplify them outside the radical sign.

6. Rational Exponents: A radical can also be expressed as a rational exponent, where $\sqrt[n]{a^m} = a^{\frac{m}{n}}$.

7. Decimal Approximation: The exact form of a radical may not be practical for numerical calculations, so we often use a decimal approximation for ease of use in real-world applications.