Simplify (6 square root of 10)/(8 square root of 6)
The question is asking you to simplify a mathematical expression that involves a fraction with radicals (square roots) in both the numerator (the top part of the fraction) and the denominator (the bottom part of the fraction). Specifically, the expression to be simplified is the fraction formed by dividing 6 times the square root of 10 by 8 times the square root of 6. The task is to carry out this simplification by applying the properties of radicals and the rules of fraction arithmetic in order to express the result in its simplest form.
$\frac{6 \sqrt{10}}{8 \sqrt{6}}$
Answer
Solution:
Step 1: Simplify the Numerator and Denominator
Step 1.1: Extract the greatest common factor from the numerator $6\sqrt{10}$.
$$\frac{2(3\sqrt{10})}{8\sqrt{6}}$$
Step 1.2: Simplify by removing common factors.
Step 1.2.1: Extract the greatest common factor from the denominator $8\sqrt{6}$.
$$\frac{2(3\sqrt{10})}{2(4\sqrt{6})}$$
Step 1.2.2: Divide out the common factor of $2$.
$$\frac{\cancel{2}(3\sqrt{10})}{\cancel{2}(4\sqrt{6})}$$
Step 1.2.3: Present the simplified fraction.
$$\frac{3\sqrt{10}}{4\sqrt{6}}$$
Step 2: Rationalize the Radical
Combine the square roots in the numerator.
$$\frac{3\sqrt{\frac{10}{6}}}{4}$$
Step 3: Simplify the Radical
Step 3.1: Factor out common factors in the radical.
$$\frac{3\sqrt{\frac{2(5)}{6}}}{4}$$
Step 3.2: Simplify by canceling common factors.
Step 3.2.1: Factor out the common factor from the denominator of the radical.
$$\frac{3\sqrt{\frac{2 \cdot 5}{2 \cdot 3}}}{4}$$
Step 3.2.2: Divide out the common factor of $2$.
$$\frac{3\sqrt{\frac{\cancel{2} \cdot 5}{\cancel{2} \cdot 3}}}{4}$$
Step 3.2.3: Present the simplified radical.
$$\frac{3\sqrt{\frac{5}{3}}}{4}$$
Step 4: Further Simplification
Step 4.1: Convert the radical to a fraction of square roots.
$$\frac{3\frac{\sqrt{5}}{\sqrt{3}}}{4}$$
Step 4.2: Rationalize the denominator by multiplying by $\frac{\sqrt{3}}{\sqrt{3}}$.
$$\frac{3\left(\frac{\sqrt{5}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}\right)}{4}$$
Step 4.3: Simplify the expression.
Step 4.3.1: Multiply the square roots in the numerator.
$$\frac{3\frac{\sqrt{5}\sqrt{3}}{\sqrt{3}\sqrt{3}}}{4}$$
Step 4.3.2: Apply the power of a power rule.
$$\frac{3\frac{\sqrt{5}\sqrt{3}}{(3^{\frac{1}{2}})^2}}{4}$$
Step 4.3.3: Simplify the exponent.
$$\frac{3\frac{\sqrt{5}\sqrt{3}}{3}}{4}$$
Step 4.4: Combine the square roots.
$$\frac{3\frac{\sqrt{15}}{3}}{4}$$
Step 5: Final Simplification
Combine the numerator.
$$\frac{\frac{3\sqrt{15}}{3}}{4}$$
Step 6: Reduce the fraction by canceling common factors.
Step 6.1: Simplify the numerator.
$$\frac{\frac{\cancel{3}\sqrt{15}}{\cancel{3}}}{4}$$
Step 6.2: Divide the square root by $1$.
$$\frac{\sqrt{15}}{4}$$
Step 7: Present the Result
- Exact Form: $$\frac{\sqrt{15}}{4}$$
- Decimal Form: $$0.96824583\ldots$$
Knowledge Notes:
To solve this problem, we utilized several mathematical concepts and rules:
Greatest Common Factor (GCF): The largest factor that divides two numbers. We used it to simplify the fraction.
Simplifying Square Roots: Combining and reducing square roots to simplify expressions.
Rationalizing the Denominator: Multiplying by a form of one to eliminate square roots from the denominator.
Power of a Power Rule: For any non-zero number $a$ and integers $m$ and $n$, $(a^m)^n = a^{mn}$.
Product Rule for Radicals: For non-negative numbers $a$ and $b$, $\sqrt{a}\sqrt{b} = \sqrt{ab}$.
Reducing Fractions: Dividing the numerator and denominator by their GCF to simplify the fraction.
Decimal Representation: Expressing the exact form as a decimal, which may be an approximation due to the irrational nature of square roots.
