Question

Simplify $ \dfrac{{{{\left( {2x} \right)}^{ - 4}}}}{{{x^{ - 1}}.x}} $ and write it using only positive exponents?

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
 $ a^m \times a^n = a^{(m + 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-1=1a
Where a is the base of the exponents.
The third law is:
 $ {a^0} = 1 $
By using these above three laws we can simplify the expression.

Complete step by step answer:
Step1:
We are given an expression $ \dfrac{{{{\left( {2x} \right)}^{ - 4}}}}{{{x^{ - 1}}.x}} $ and we have to find its value. First we will solve the denominator and find the product by using the law of exponent:
 $ a^m \times a^n = a^{(m + n)} $
Here \[a = x\]; \[m = - 1\]and \[n = 1\]
On application of law we will get:
 $ \Rightarrow \dfrac{{{{\left( {2x} \right)}^{ - 4}}}}{{{x^0}}} $

Step2:
Now we will use the property of exponents that $ {a^0} = 1 $ . Here $ a = x $ we will get:
 $ \Rightarrow \dfrac{{{{\left( {2x} \right)}^{ - 4}}}}{1} $
Step3: On solving for the numerator we will use the property of exponents that a-1=1a
Here $ a = 2x $ on applying the property we will get:
 $ \Rightarrow \dfrac{1}{{{{\left( {2x} \right)}^4}}} $
On expanding:
 $ \Rightarrow \dfrac{1}{{2x \times 2x \times 2x \times 2x}} $
On further expanding the number we will get:
 $ \Rightarrow \dfrac{1}{{2 \times 2 \times 2 \times 2 \times x \times x \times x \times x}} $
On multiplication of all $ x $ together and 2 together we will get:
 $ \Rightarrow \dfrac{1}{{16{x^4}}} $
As we have to write in positive exponents so we will keep it in fractional form.

Note: 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.