Question

Which of the following numbers is the largest?
A) \[{{\left( 1+{{2}^{\dfrac{1}{2}}} \right)}^{2}}\]
B) \[2{{\left( \dfrac{1}{6}+\dfrac{1}{3} \right)}^{-1}}\]
C) \[{{24}^{0.5}}\]
D) \[\left( \dfrac{10}{7} \right)\left( \dfrac{9}{6} \right)\left( \dfrac{8}{5} \right)\]

Answer 1

Hint: In the given question we have been asked to find the largest expression among the 4 options. In order to find the largest option, we will need to solve each expression one by one using the laws of exponent and powers and simplification using mathematical options such as addition, subtraction, multiplication and division.

Complete step-by-step solution:
A) We have given that,
\[{{\left( 1+{{2}^{\dfrac{1}{2}}} \right)}^{2}}\]
Expanding the above using the identity\[{{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab\], we get
\[\left( {{1}^{2}}+2\left( 1 \right)\left( {{2}^{\dfrac{1}{2}}} \right)+{{\left( {{2}^{\dfrac{1}{2}}} \right)}^{2}} \right)\]
Solving the number in the above equation, we will get
\[\left( 1+{{2}^{1}}\left( {{2}^{\dfrac{1}{2}}} \right)+2 \right)\]
Using the laws of exponent and power i.e. \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\]
We obtained,
\[\left( 1+{{2}^{\dfrac{3}{2}}}+2 \right)=3+{{2}^{\dfrac{3}{2}}}=3+\left( 2\times 1.41 \right)=3+2.82=5.82\]
Thus,
\[{{\left( 1+{{2}^{\dfrac{1}{2}}} \right)}^{2}}=5.82\]

B) We have given that,
\[2{{\left( \dfrac{1}{6}+\dfrac{1}{3} \right)}^{-1}}\]
Solving the bracket by taking the LCM, we get
\[2{{\left( \dfrac{3}{6} \right)}^{-1}}=2{{\left( \dfrac{1}{2} \right)}^{-1}}\]
Using the negative exponent property i.e. \[{{a}^{-m}}=\dfrac{1}{{{a}^{m}}}\]
\[2\left( 2 \right)=4\]
Thus,
\[2{{\left( \dfrac{1}{6}+\dfrac{1}{3} \right)}^{-1}}=4\]

C) We have given that,
\[{{24}^{0.5}}\]
As we know that,
\[0.5=\dfrac{1}{2}\]
Now,
\[{{24}^{0.5}}={{\left( 24 \right)}^{\dfrac{1}{2}}}=\sqrt{24}=2\sqrt{6}=2\times 2.45=4.9\]
Thus,
\[{{24}^{0.5}}=4.9\]

D) We have given that,
\[\left( \dfrac{10}{7} \right)\left( \dfrac{9}{6} \right)\left( \dfrac{8}{5} \right)\]
Writing the above expression in the form of factors,
Such that,
\[10=2\times 5\]
\[9=3\times 3\]
\[6=2\times 3\]
\[8=2\times 4\]
Therefore,
The given expression can be rewritten as,
\[\left( \dfrac{2\times 5}{7} \right)\times \left( \dfrac{3\times 3}{2\times 3} \right)\times \left( \dfrac{2\times 4}{5} \right)\]
Cancelling out all the common terms, we get
\[\left( \dfrac{1}{7} \right)\left( \dfrac{3}{1} \right)\left( \dfrac{8}{1} \right)=\dfrac{3\times 8}{7}=\dfrac{24}{7}=3.42\]
Thus,
\[\left( \dfrac{10}{7} \right)\left( \dfrac{9}{6} \right)\left( \dfrac{8}{5} \right)=3.42\]

Hence, the option (A) has the largest number and this is the correct answer.


Note: While solving these types of questions, students should be very careful while during the calculation parts to avoid making any type of errors or mistakes. They should need to know about the concepts of using the mathematical operations such as addition, subtraction, multiplication and division. They should also keep in mind the laws of exponent and power as many parts have been solved or simplified using the properties of exponent and powers. You should be very explicitly noted all the digits because even a single wrongly noted digit will obtain the wrong answer.