Question

How do you evaluate the expression $\dfrac{{{{\left( { - 4} \right)}^2}{{\left( { - 4} \right)}^{ - 3}}}}{{{{\left( { - 4} \right)}^{ - 5}}}}$ using the properties?

Answer 1

Hint: We are given an expression and we have to solve it using the laws of exponents. Then to solve the expression which we're given we will use the following laws of exponents
am×an=am+n
If ‘a’ is a non-zero integer or a non-zero rational number and m and n are positive integers, then
This can be solved and the bases are the same.
The second law is:
$a^m÷a^n =a^m.a^n=a^{-(n-m)}$
Where m and n are whole numbers and mWe can generalize that if ‘a’ is a non-zero integer or a non-zero rational number and m and n are positive integers, such that m>n, then
$a^m÷a^n =a^{m-n}$
By using these above two laws we can use the value of the expression.

Complete step-by-step answer:
Step1: We are given an expression $\dfrac{{{{\left( { - 4} \right)}^2}{{\left( { - 4} \right)}^{ - 3}}}}{{{{\left( { - 4} \right)}^{ - 5}}}}$ and we have to find its value. First we will solve the numerator and find the product by using the law of exponent:
$a^m×a^n=a^{m+n}$
Here\[a = - 4\]; \[m = 2\] and \[n = - 3\]
On application of law we will get:
$ \Rightarrow \dfrac{{{{\left( { - 4} \right)}^2}{{\left( { - 4} \right)}^{ - 3}}}}{{{{\left( { - 4} \right)}^{ - 5}}}}$
$ \Rightarrow \dfrac{{{{\left( { - 4} \right)}^{2 + ( - 3)}}}}{{{{\left( { - 4} \right)}^{ - 5}}}}$
On solving the powers we will get:
$ \Rightarrow \dfrac{{{{\left( { - 4} \right)}^{ - 1}}}}{{{{\left( { - 4} \right)}^{ - 5}}}}$
Step2: Now we will apply the another law of exponent i.e. $a^m÷a^n=a^{m-n}$ when m>n
Here $a = - 4;m = - 1;n = - 5$
On applying the law using these values we will get:
$ \Rightarrow \dfrac{{{{\left( { - 4} \right)}^{ - 1}}}}{{{{\left( { - 4} \right)}^{ - 5}}}} = {\left( 4 \right)^{ - 1 - ( - 5)}}$
$ \Rightarrow {\left( { - 4} \right)^{ - 1 + 5}}$
On solving the expression we will get:
$ = {\left( { - 4} \right)^4}$
As the power is even so it will remove the negative sign and we will get:
On expanding:
$ \Rightarrow {\left( { - 1} \right)^4}{\left( 4 \right)^4}$
$ = {\left( 4 \right)^4}$
On further expanding the number we will get:
$ = 4 \times 4 \times 4 \times 4$
On multiplication we will get:
$ = 256$

Hence the answer is $256$

Note:
In this type of questions of exponents are solved by the laws of exponent and it's not only these two laws there are many others also. We have to only apply the laws correctly in case of negative power. Students mainly get confused in case of negative power numbers get reciprocal to make the power positive. Small calculations are required, no big calculations are there. Keep in mind that change of sign. On opening the bracket with a minus sign that on opening with it signs will get reversed. Then the question will not get wrong.