Question

How do you simplify \[\left( {\dfrac{1}{2}} \right) \div \left( {\dfrac{{\sqrt 3 }}{2}} \right)\]?

Answer 1

Hint: Dividing two fractions is the same as multiplying the first fraction by the reciprocal of the second fraction. The first step to divide fractions is to find the reciprocal (reverse the numerator and denominator) of the second fraction. Next, multiply the two numerators. Then, multiply the two denominators.

Complete step-by-step solution:
Let us write the given data,
\[\Rightarrow \left( {\dfrac{1}{2}} \right) \div \left( {\dfrac{{\sqrt 3 }}{2}} \right)\]
To divide these two fractions, change division to multiplication and
multiply by the reciprocal of the divisor.
\[\Rightarrow \left( {\dfrac{1}{2}} \right) \div \left( {\dfrac{{\sqrt 3 }}{2}} \right)\] = \[\dfrac{1}{2} \times \dfrac{2}{{\sqrt 3 }}\]
Simplifying the terms, we get
\[\Rightarrow \left( {\dfrac{1}{2}} \right) \div \left( {\dfrac{{\sqrt 3 }}{2}} \right)\] = \[\dfrac{1}{{\sqrt 3 }}\]

Hence the correct answer is \[\dfrac{1}{{\sqrt 3 }}\]

Additional information: When dividing radicals (with the same index), divide under the radical, and then divide in front of the radical (divide any values multiplied times the radicals). Divide out front and divide under the radicals. Then simplify the result. You have just "rationalized" the denominator.
Product Raised to a Power Rule: This rule states that the product of two or more numbers raised to a power is equal to the product of each number raised to the same power. The Product Raised to a Power Rule is important because you can use it to multiply radical expressions. Note that the roots are the same—you can combine square roots with square roots, or cube roots with cube roots, for example. But you can’t multiply a square root and a cube root using this rule.

Note: The key point is to flip the divisor into a reciprocal and change the division sign into a multiplication sign and multiply then simplify if possible. We can divide fractions by whole numbers by multiplying the bottom of the fraction by the whole number. To divide, if the given question consists of mixed numbers then: change each mixed number to an improper fraction and multiply by the reciprocal of the divisor, simplifying if possible.