Question

Calculate the value of the following: \[\sqrt{\dfrac{25}{32}\times 2\dfrac{13}{18}\times 0.25}\]?

Answer 1

Hint: To evaluate the value of the given expression, we should know how to perform multiplication and division of fractions. To do multiplication, we need to express numerators and denominators of both in factored form. We should also know how to convert improper fractions to proper fractions. The improper fractions of the form \[a\dfrac{b}{c}\] can be written in proper form as \[\dfrac{ac+b}{c}\]. To convert decimals to fractions, we multiply its numerator and denominator by 10 raised to the number of numbers after the decimal point.

Complete step by step answer:
We are asked to calculate \[\sqrt{\dfrac{25}{32}\times 2\dfrac{13}{18}\times 0.25}\]. To perform this calculation, we first have to convert the improper fraction and decimal to the proper fraction and write them as their factored form.
The improper fraction given is \[2\dfrac{13}{18}\]. We can write this in proper form as \[\dfrac{18\times 2+13}{18}\]. Simplifying this, we get \[\dfrac{49}{18}\]. The numerator and denominator are 49 and 18 respectively. We can write them in factored form as \[7\times 7\And 2\times 3\times 3\] respectively.
The decimal given is \[0.25\], it has 2 digits after the decimal point. To write it as the proper fraction, we have to multiply it by \[\dfrac{100}{100}\]. By doing this, we get \[\dfrac{25}{100}\] writing in factored form and canceling out common factors, we get \[\dfrac{1}{4}\].
Writing all terms in their factored form, we get \[\sqrt{\dfrac{5\times 5}{2\times 2\times 2\times 2\times 2}\times \dfrac{7\times 7}{2\times 3\times 3}\times \dfrac{1}{2\times 2}}\]. Grouping the similar terms and taking proper power out of the square root, we get \[\dfrac{5\times 7}{{{2}^{4}}\times 3}\]. Simplifying it we get \[\dfrac{35}{48}\].

Note:
These types of questions can be simply converting all terms to proper fractions and writing them in their factored form. After writing the factored form, we cancel out the common factors that terms have in common in their numerator and denominator. Calculation mistakes while solving this should be avoided.