Question

How do you simplify \[{\left( {\dfrac{3}{7}} \right)^{ - 2}}\] ?

Answer 1

Hint: We can see this problem is from indices and powers. We know that if we have \[{a^{ - n}}\] then we can write it as \[\dfrac{1}{{{a^n}}}\] . Using this we will get a fraction. We can simplify the fraction to get a required answer. Since we have exponent power 2 will have a square of a number.

Complete step-by-step answer:
Given, \[{\left( {\dfrac{3}{7}} \right)^{ - 2}}\] .
Since we know \[{a^{ - n}} = \dfrac{1}{{{a^n}}}\] . So above becomes
 \[{\left( {\dfrac{3}{7}} \right)^{ - 2}} = \dfrac{1}{{{{\left( {\dfrac{3}{7}} \right)}^2}}}\]
 \[ = \dfrac{1}{{\left( {\dfrac{{{3^2}}}{{{7^2}}}} \right)}}\]
This can be written as,
 \[ = \dfrac{{{7^2}}}{{{3^2}}}\]
 \[ = \dfrac{{49}}{9}\] . We can stop it here.
We can put it in decimal form as 5.45
So, the correct answer is “ 5.45”.

Note: We know the laws of indices.
Rule 1: if a constant or variable has index as ‘0’, then the result will be equal to one, regardless of any base value. That is \[{a^0} = 1\] and ‘a’ is not equal to zero.
Rule 2: If the index is a negative value, then it can be shown as the reciprocal of the positive index raised to the same variable. That is \[{a^{ - 1}} = \dfrac{1}{a}\] .
Rule 3: If we have two variables with the same base and different exponent, we add the exponent that is \[{x^a} \times {x^b} = {x^{a + b}}\] .
Rule 4: If we have division of two number with same base, that is \[\dfrac{{{x^a}}}{{{x^b}}} = {x^{a - b}}\] .
These are some basic laws.