Question

Simplify the following: \[{\left( {\dfrac{{81}}{{16}}} \right)^{8/4}} \times \left[ {{{\left( {\dfrac{{25}}{4}} \right)}^{ - 3/2}} \div {{\left( {\dfrac{5}{2}} \right)}^{ - 3}}} \right]\].

Answer 1

Hint: This solution will require the concepts of Exponents and simplification. First express the terms in the brackets as exponents of the simplest number possible. Then simplify and follow the BODMAS rule further. The BODMAS rule stands for Bracket Of Division Multiplication Addition Subtraction, i.e. all these operations need to be carried out in the mentioned order.

Complete step by step solution:
We are given to simplify \[{\left( {\dfrac{{81}}{{16}}} \right)^{8/4}} \times \left[ {{{\left( {\dfrac{{25}}{4}} \right)}^{ - 3/2}} \div {{\left( {\dfrac{5}{2}} \right)}^{ - 3}}} \right]\].
First express the fractions in the brackets as the exponents of the simplest number possible, observe that all the numbers in the brackets can be expressed as exponents of \[2,3\] and \[5\]:
       \[{\left( {\dfrac{{81}}{{16}}} \right)^{8/4}} \times \left[ {{{\left( {\dfrac{{25}}{4}} \right)}^{ - 3/2}} \div {{\left( {\dfrac{5}{2}} \right)}^{ - 3}}} \right]\] \[ = \] \[{\left( {\dfrac{{{3^4}}}{{{2^4}}}} \right)^{8/4}} \times \left[ {{{\left( {\dfrac{{{5^2}}}{{{2^2}}}} \right)}^{ - 3/2}} \div {{\left( {\dfrac{5}{2}} \right)}^{ - 3}}} \right]\]
\[ \Rightarrow \]\[{\left( {\dfrac{{81}}{{16}}} \right)^{8/4}} \times \left[ {{{\left( {\dfrac{{25}}{4}} \right)}^{ - 3/2}} \div {{\left( {\dfrac{5}{2}} \right)}^{ - 3}}} \right]\] \[ = \] \[{\left[ {{{\left( {\dfrac{3}{2}} \right)}^4}} \right]^{8/4}} \times \left[ {{{\left[ {{{\left( {\dfrac{5}{2}} \right)}^2}} \right]}^{ - 3/2}} \div {{\left( {\dfrac{5}{2}} \right)}^{ - 3}}} \right]\]
Multiplying the powers as we know \[{\left[ {{{\left( a \right)}^b}} \right]^c} = {\left( a \right)^{bc}}\]:
\[ \Rightarrow \]\[{\left( {\dfrac{{81}}{{16}}} \right)^{8/4}} \times \left[ {{{\left( {\dfrac{{25}}{4}} \right)}^{ - 3/2}} \div {{\left( {\dfrac{5}{2}} \right)}^{ - 3}}} \right]\] \[ = \] \[{\left( {\dfrac{3}{2}} \right)^8} \times \left[ {{{\left( {\dfrac{5}{2}} \right)}^{ - 3}} \div {{\left( {\dfrac{5}{2}} \right)}^{ - 3}}} \right]\]
Now we simplify following the BODMAS rule. So first we have already simplified the brackets next we have to do the division:
\[ \Rightarrow \]\[{\left( {\dfrac{{81}}{{16}}} \right)^{8/4}} \times \left[ {{{\left( {\dfrac{{25}}{4}} \right)}^{ - 3/2}} \div {{\left( {\dfrac{5}{2}} \right)}^{ - 3}}} \right]\] \[ = \] \[{\left( {\dfrac{3}{2}} \right)^8} \times \left[ 1 \right]\]
\[ \Rightarrow \]\[{\left( {\dfrac{{81}}{{16}}} \right)^{8/4}} \times \left[ {{{\left( {\dfrac{{25}}{4}} \right)}^{ - 3/2}} \div {{\left( {\dfrac{5}{2}} \right)}^{ - 3}}} \right]\] \[ = \] \[{\left( {\dfrac{3}{2}} \right)^8}\].

Hence, \[{\left( {\dfrac{3}{2}} \right)^8}\] is the required answer.

Note:
For solving these types of problems students must be familiar with the rules of exponents and the BODMAS rule. Note that the BODMAS rule is often referred to as the PEDMAS rule which stands for Parenthesis Exponents Division Multiplication Addition Subtraction. These rules help in solving complex problems involving multiple operators. These are acronyms which help in remembering the order in which the operations need to be performed. In these problems the calculations must be done very carefully.