Question

Write $ {a^{1.5}} $ in the radical form.

Answer 1

Hint: In this problem, we have given an expression and we have to convert it into radical form. An expression is said to be in radical form if the expression have a radical sign $ \left( {\sqrt {} } \right) $ and the simplest radical form is that when the number inside the radical sign is indivisible by perfect square and only divisible by $ 1 $ .
Formula used: Rule for exponents-
1. $ {\left( {{x^a}} \right)^b} = {x^{a \times b}} $ and $ {x^{a \times b}} = {\left( {{x^a}} \right)^b} $
2. $ {x^{\dfrac{1}{n}}} = \sqrt[n]{x} $

Complete step-by-step answer:
To find the value of $ 1.5 $ in a fraction, we need to multiply the numerator and denominator by $ 10 $ and after simplifying we get the value $ \dfrac{3}{2} $ and to modify the expression we have used the rules for exponents and the inverse of a power is termed as root.

Now, we have to convert the expression in radical form, we know that, $ 1.5 = \dfrac{3}{2} $ and now, we can write the expression as, $ {a^{\dfrac{3}{2}}} $ and now, we can also write this expression as $ {a^{3 \times \dfrac{1}{2}}} $ . And this $ ({a^{3 \times \dfrac{1}{2}}}) $ can also be written, by using the above first rule, as $ {\left( {{a^3}} \right)^{\dfrac{1}{2}}} $ . Now, we can write the expression by using the above second rule as: $ \sqrt[2]{{{a^3}}} $ and we know that, a square root is same as a $ \dfrac{1}{2} $ exponent, then our expression will become
 $ \sqrt {{a^3}} $ and this is the radical form of $ {a^{1.5}} $ .
So, the correct answer is “ $ \sqrt {{a^3}} $ ”.

Note: The number of times a number is multiplied with itself is termed as exponent. For example, $ 2 \times 2 \times 2 = {2^3} $ , here $ 2 $ is multiplied with itself three times then the exponent of $ 2 $ becomes $ 3 $ . In this example, $ 2 $ is the base and $ 3 $ is the power, we will pronounce it as $ 2 $ raised to the power $ 3 $ .