What is the value of the rationalizing factor of $2\sqrt[3]{5}$ ? (a) $\sqrt[3]{5}$ (b) $\sqrt[3]{{{5}^{2}}}$ (c) ${{5}^{2}}$ (d) ${{5}^{3}}$
ANSWER
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Hint: Here, we will use the definition of rationalizing factor to find the rationalizing factor of the given number. The rationalizing factor of a number is the number by which it must be multiplied so that it becomes rational.
Complete Step-by-Step solution: When the denominator of an expression is a surd which can be reduced to an expression with rational denominator, this process is known as rationalizing the denominator of the surd. If a surd is present in the denominator of an equation, then to simplify it or to omit the surds from the denominator, rationalization of surds is used. Surds are irrational numbers but, if we multiply a surd with a suitable factor the result of the multiplication will be a rational number. This is the basic principle involved in rationalizing surds. The factor by which rationalization is done is called rationalizing factor. If the product of two surds is a rational number, then each surd is a rationalizing factor to each other. In other words, the process of reducing a given surd to a rational form after multiplying it by a suitable surd is known as rationalization. Since, the number given to us is: $2\sqrt[3]{5}=2{{\left( 5 \right)}^{\dfrac{1}{3}}}$ This given number is surd. If we multiply this given surd or this given irrational number by ${{\left( 5 \right)}^{\dfrac{2}{3}}}$ , we get: \[\begin{align} & 2{{\left( 5 \right)}^{\dfrac{1}{3}}}\times {{\left( 5 \right)}^{\dfrac{2}{3}}} \\ & =2\times {{\left( 5 \right)}^{\dfrac{\left( 1+2 \right)}{3}}} \\ & =2\times {{\left( 5 \right)}^{\dfrac{3}{3}}} \\ & =2\times 5 \\ & =10 \\ \end{align}\] Since, 10 is a rational number. So, ${{\left( 5 \right)}^{\dfrac{2}{3}}}=3\sqrt{{{5}^{2}}}$ is the rationalizing factor of $2\sqrt[3]{5}$ . Hence, option (b) is the correct answer.
Note: Students should note that the rationalizing factor should be the smallest number which when multiplied by the given number gives a rational number.