Question

Simplify the following expression: $\left[ {{{\left( {\dfrac{{ - 3}}{4}} \right)}^3} - {{\left( {\dfrac{{ - 5}}{2}} \right)}^3}} \right] \times {\left( { - \dfrac{2}{3}} \right)^4}$

Answer 1

Hint: According to given in the question we have to find the value or simplify the given expression $\left[ {{{\left( {\dfrac{{ - 3}}{4}} \right)}^3} - {{\left( {\dfrac{{ - 5}}{2}} \right)}^3}} \right] \times {\left( { - \dfrac{2}{3}} \right)^4}$so, first of all we have to use the BODMAS rule as explained below:
B – Brackets of (according to this first of all we have solve the smaller bracket then we have to solve curly bracket and after it we have to solve large bracket)
O- Orders
D – Divide
M – Multiply
A – Subtract
A – Addition
Hence, with the help of the BODMAS rule as explained above first of all we have to solve the terms inside the smaller bracket and in the smaller bracket first of all we will divide and then subtract the terms inside the smaller bracket. Now we will solve the value inside the curly bracket so first we will divide and then subtract and after it we have to solve all the terms inside the large bracket to find the value of the given expression. Now, we have to solve the terms inside smaller, and large brackets with help of the formula as given below:

Formula used: \[ \Rightarrow {(a)^3} = a \times a \times a...............(1)\]
$ \Rightarrow {(a)^4} = a \times a \times a \times a............(2)$
Now, we will solve each term of the expression separately with the help of the formulas (1) and (2) and then we will put all the values in the given expression to find the value or to simplify the given expression.

Complete step-by-step answer:
Given Expression: $\left[ {{{\left( {\dfrac{{ - 3}}{4}} \right)}^3} - {{\left( {\dfrac{{ - 5}}{2}} \right)}^3}} \right] \times {\left( { - \dfrac{2}{3}} \right)^4}$
Step 1: First of all we have to solve the terms inside the brackets with the help of the BODMAS rule as mentioned in the solution hint. So, we have to solve the terms inside the smaller bracket with the help of the formula (12) as mentioned in the solution hint.
$
   \Rightarrow {\left( { - \dfrac{2}{3}} \right)^4} = - \dfrac{2}{3} \times - \dfrac{2}{3} \times - \dfrac{2}{3} \times - \dfrac{2}{3} \\
   \Rightarrow {\left( { - \dfrac{2}{3}} \right)^4} = \dfrac{{16}}{{81}} \\
 $
Step 2: Now, we have to solve another term in the given expression with the help of the formula (1) as mentioned in the solution hint.
\[
   \Rightarrow {\left( { - \dfrac{3}{4}} \right)^3} = \left( { - \dfrac{3}{4}} \right) \times \left( { - \dfrac{3}{4}} \right) \times \left( { - \dfrac{3}{4}} \right) \\
   \Rightarrow {\left( { - \dfrac{3}{4}} \right)^3} = - \dfrac{{27}}{{64}} \\
 \]
Step 3: Now, same as we have to solve another term in the given expression with the help of the formula (1) as mentioned in the solution hint.
\[
   \Rightarrow {\left( { - \dfrac{5}{2}} \right)^3} = \left( { - \dfrac{5}{2}} \right) \times \left( { - \dfrac{5}{2}} \right) \times \left( { - \dfrac{5}{2}} \right) \\
   \Rightarrow {\left( { - \dfrac{5}{2}} \right)^3} = - \dfrac{{125}}{8} \\
 \]
Step 4: Now, we have to substitute all the values of the terms in the given expression (1) to simplify it.
$ = \left[ {\left( { - \dfrac{{27}}{{64}}} \right) - \left( { - \dfrac{{125}}{8}} \right)} \right] \times \dfrac{{16}}{{81}}$
Now, on solving the obtained expression with the help of BODMAS rule as mentioned in the solution hint.
$ = \left[ { - \dfrac{{27}}{{64}} + \dfrac{{125}}{8}} \right] \times \dfrac{{16}}{{81}}$
Now, to solve the obtained expression we have to find the L.C.M. of the terms inside the larger bracket.
$
   = \left[ {\dfrac{{ - 27 + 1000}}{{64}}} \right] \times \dfrac{{16}}{{81}} \\
   = - \dfrac{{973}}{{64}} \times \dfrac{{16}}{{81}} \\
 $
Now, on solving the obtained expression just above,
$ = - \dfrac{{973}}{{324}}$

Hence, with the help of the formulas (1) and (2) and BODMAS rule we have simplified the given expression: $\left[ {{{\left( {\dfrac{{ - 3}}{4}} \right)}^3} - {{\left( {\dfrac{{ - 5}}{2}} \right)}^3}} \right] \times {\left( { - \dfrac{2}{3}} \right)^4}$$ = - \dfrac{{973}}{{324}}$

Note: If a negative number is multiplied with a negative number then the number will become positive but if a negative number is multiplied with a positive number then the number will become negative.
It is easy to solve expressions by finding the values of all it’s parts separately and after finding the values of all the terms we can substitute the terms in the expression to solve it.