simplify the square root of 7 over 18
The Correct Answer and Explanation is:
To simplify the expression 718\sqrt{\frac{7}{18}}187, follow these steps:
Step 1: Apply the property of square roots to a fraction
718=718\sqrt{\frac{7}{18}} = \frac{\sqrt{7}}{\sqrt{18}}187=187
Step 2: Simplify the denominator 18\sqrt{18}18
18 is not a perfect square, but it can be factored as 9×29 \times 29×2. Since 9 is a perfect square:18=9⋅2=9⋅2=32\sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2}18=9⋅2=9⋅2=32
Step 3: Substitute back into the expression
718=732\frac{\sqrt{7}}{\sqrt{18}} = \frac{\sqrt{7}}{3\sqrt{2}}187=327
Step 4: Rationalize the denominator
To eliminate the square root from the denominator, multiply both the numerator and denominator by 2\sqrt{2}2:732⋅22=143⋅2=146\frac{\sqrt{7}}{3\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{14}}{3 \cdot 2} = \frac{\sqrt{14}}{6}327⋅22=3⋅214=614
Final Simplified Answer:
146\boxed{\frac{\sqrt{14}}{6}}614
Explanation
Simplifying square roots involving fractions follows a specific process based on the properties of radicals. In this case, we begin with the square root of the fraction 718\frac{7}{18}187. The square root of a fraction can be split into the square root of the numerator over the square root of the denominator, which gives 718\frac{\sqrt{7}}{\sqrt{18}}187.
The next step is to simplify the denominator. Although 18 is not a perfect square, it can be factored into 9 times 2. Since 9 is a perfect square, its square root is 3. This means 18=32\sqrt{18} = 3\sqrt{2}18=32. Substituting this simplified form into the expression gives 732\frac{\sqrt{7}}{3\sqrt{2}}327.
At this stage, the denominator still contains a radical, which is generally not preferred in final mathematical answers. To resolve this, we use a process called rationalizing the denominator. This involves multiplying the entire expression by a form of 1 that will eliminate the radical in the denominator. In this case, we multiply by 22\frac{\sqrt{2}}{\sqrt{2}}22, which is equal to 1 and does not change the value of the expression. The result becomes 146\frac{\sqrt{14}}{6}614, since 7⋅2=14\sqrt{7} \cdot \sqrt{2} = \sqrt{14}7⋅2=14, and 32⋅2=3⋅2=63\sqrt{2} \cdot \sqrt{2} = 3 \cdot 2 = 632⋅2=3⋅2=6.
The final expression, 146\frac{\sqrt{14}}{6}614, is simplified and rationalized. This result cannot be simplified further since 14 is not a perfect square and the square root of 14 cannot be expressed in simpler radical form.
