Question

Evaluate:
(i) Find the value of $ {({x^4})^2} \div \left( {{x^6} \times \dfrac{1}{{{x^2}}}} \right) $ for $ x = 2 $ .
(ii) Find the value of $ {p^5} \times {p^3} \div {\left( {{p^3}} \right)^2} $ for $ p = - 1 $ .
(iii) Find the value of $ {({a^5} \times {a^2})^2} \times {a^2} \div {\left( {{a^2}} \right)^3} $ for $ a = - 2 $ .

Answer 1

Hint: First we will substitute the given values in the expression. Then evaluate the powers accordingly and open the brackets and solve according to the PEMDAS rule. Also, state the PEMDAS rule.

Complete step by step solution:
I.
We will start off by stating the PEMDAS rule. According to the PEMDAS rule, parenthesis is opened first then we will evaluate the exponents that are powers and square roots. After that we solve the multiplications and division and then finally we evaluate the addition and subtraction operation.
Now we start by substituting value in the expression.
 $
   = {({x^4})^2} \div \left( {{x^6} \times \dfrac{1}{{{x^2}}}} \right) \\
   = {({2^4})^2} \div \left( {{2^6} \times \dfrac{1}{{{2^2}}}} \right) \;
  $
Now here we start applying the PEMDAS rule, and start evaluating the exponent values.
 $
   = {({2^4})^2} \div \left( {{2^6} \times \dfrac{1}{{{2^2}}}} \right) \\
   = {(16)^2} \div \left( {64 \times \dfrac{1}{4}} \right) \\
  $
Here, we will evaluate the multiplication and division.
 $
   = {(16)^2} \div \left( {16} \right) \\
   = 16 \;
 $

II.Now we start by substituting value in the expression.
 $
   = {p^5} \times {p^3} \div {\left( {{p^3}} \right)^2} \\
   = {( - 1)^5} \times {( - 1)^3} \div {\left( {{{( - 1)}^3}} \right)^2} \\
  $
Now here we start applying the PEMDAS rule, and start evaluating the exponent values.
 $
   = {( - 1)^5} \times {( - 1)^3} \div {\left( {{{( - 1)}^3}} \right)^2} \\
   = ( - 1) \times ( - 1) \div {\left( { - 1} \right)^2} \\
  $
Here, we will evaluate the multiplication and division.
 $
   = ( - 1) \times ( - 1) \div {\left( { - 1} \right)^2} \\
   = ( - 1) \times ( - 1) \div 1 \\
   = 1 \;
  $

III.Now we start by substituting value in the expression.
 $
   = {({a^5} \times {a^2})^2} \times {a^2} \div {\left( {{a^2}} \right)^3} \\
   = {({( - 2)^5} \times {( - 2)^2})^2} \times {( - 2)^2} \div {\left( {{{( - 2)}^2}} \right)^3} \\
  $
Now here we start applying the PEMDAS rule, and start evaluating the exponent values.
 $
   = {({( - 2)^5} \times {( - 2)^2})^2} \times {( - 2)^2} \div {\left( {{{( - 2)}^2}} \right)^3} \\
   = {(( - 32) \times 4)^2} \times {(4)^2} \div {\left( 4 \right)^3} \\
   = {(128)^2} \times 16 \div 64 \\
  $
Here, we will evaluate the multiplication and division.
 $
   = {(128)^2} \times 16 \div 64 \\
   = 16384 \times 16 \div 64 \\
   = 16384 \div 4 \\
   = 4096 \;
  $

Note: The PEMDAS rule is a set of rules that prioritise the order of calculations, that is, which operation to perform first. Otherwise, it is possible to get multiple or different answers. To decide when to multiply or divide, always perform the one which appears first from left to right.
Do not solve all the equations simultaneously. Solve all the equations separately, so that you don’t miss any term of the solution. Substitute values along with their respective signs. While applying the rule, choose the operation according to the order of the rule.