Question

The ratio of \[{{4}^{3.5}}:{{2}^{5}}\] is same as:
(a) 2 : 1
(b) 4 : 1
(c) 7 : 5
(d) 5 : 2

Answer 1

Hint: In this question, we first need to simplify the decimal term in the power of 4 and then convert it accordingly. Then after simplification we get both terms in powers of 4 which gives us the result.

Complete step-by-step answer:

POWER AND INDEX:
If a number a is multiplied by itself n times, then the product is called nth power of a and is written as \[{{a}^{n}}\]. In \[{{a}^{n}}\], a is called the base and n is called the index.
Let a be a non-zero rational number and \[\dfrac{p}{q}\]be a positive rational number, then \[{{a}^{\dfrac{p}{q}}}\] may be defined as
\[{{a}^{\dfrac{p}{q}}}={{\left( {{a}^{p}} \right)}^{\dfrac{1}{q}}}\] read as qth root of pth power of a
\[{{a}^{\dfrac{p}{q}}}={{\left( {{a}^{\dfrac{1}{q}}} \right)}^{p}}={{\left( \sqrt[q]{a} \right)}^{p}}\] read as pth power of qth root of a
Now, from the given question we have,
\[\Rightarrow {{4}^{3.5}}:{{2}^{5}}\]
Let us now further rewrite the term \[{{4}^{3.5}}\]
Now, let us convert the index in that number into a fraction.
\[\Rightarrow {{4}^{3.5}}={{4}^{\dfrac{35}{10}}}\]
Now, by cancelling out the common terms in the numerator and denominator of the index we can further write it as
\[\Rightarrow {{4}^{3.5}}={{4}^{\dfrac{7}{2}}}\]
Now, by using the formulae mentioned above we can convert it as
As we already know that
\[{{a}^{\dfrac{p}{q}}}={{\left( \sqrt[q]{a} \right)}^{p}}\]
Now, on substituting the respective values in the above formula we get,
\[\Rightarrow {{4}^{3.5}}={{\left( \sqrt{4} \right)}^{7}}\]
Now, this can be further simplified and written as
\[\Rightarrow {{4}^{3.5}}={{2}^{7}}\text{ }\]
Now, by substituting this value back in the equation given in the question we get,
\[\Rightarrow {{4}^{3.5}}:{{2}^{5}}={{2}^{7}}:{{2}^{5}}\]
Now, by cancelling out the common terms we get,
\[\Rightarrow {{4}^{3.5}}:{{2}^{5}}={{2}^{2}}:1\]
Now, this can be further written as
\[\begin{align}
  & \Rightarrow {{4}^{3.5}}:{{2}^{5}}=4:1 \\
 & \text{ } \\
\end{align}\]
Hence, the correct option is (b).

Note: It is important to note that in order to simplify the given decimal index we first need to convert into the fractional form. So, that by using the corresponding formula we can further simplify it and write it in terms of the given denominator in the question.
While simplifying the terms we should be careful about the powers because neglecting them or writing other terms in that place change the corresponding value and so the final result.