Question

If \[a = 3\] and \[b = - 2\] , find the value of : \[{a^a} + {b^b}\]

Answer 1

Hint: Here, we have to find the value of \[{a^a} + {b^b}\]. We will substitute the values of \[a\] and \[b\] in the given expression. Then we will apply the exponent on the terms and simplify each term. We will then add the number to find the value of the given expression.
Formula used:
When \[a\] is raised to the power \[ - n\], it is given by the exponential formula \[{a^{ - n}} = \dfrac{1}{{{a^n}}}\]

Complete step-by-step answer:
We will substitute the values of \[a\] and \[b\] to find the value of \[{a^a} + {b^b}\].
\[{a^a} + {b^b} = {3^3} + {( - 2)^{ - 2}}\]
When \[a\] is raised to the power \[ - n\], it is given by the exponential formula \[{a^{ - n}} = \dfrac{1}{{{a^n}}}\]
By using the exponential formula, we get
\[ \Rightarrow {a^a} + {b^b} = {3^3} + {\left( {\dfrac{1}{{ - 2}}} \right)^2}\]
Now we know that the cube of 3 is 9 and square of 2 is 4. Therefore, we get
\[ \Rightarrow {a^a} + {b^b} = 27 + \left( {\dfrac{1}{4}} \right)\]
Here we will be taking Least Common Multiple for the denominators \[1\] and \[4\] , we have L.C.M as \[4\].
L.C.M \[(1,4) = 1 \times 4 = 4\]
Now, multiplying the given fractions to get the denominators equal to 4, we have
\[ \Rightarrow {a^a} + {b^b} = 27 \times \dfrac{4}{4} + \dfrac{1}{4} \times \dfrac{1}{1}\]
Now, the unlike fraction has turned into a like fraction.
\[ \Rightarrow {a^a} + {b^b} = \dfrac{{108}}{4} + \dfrac{1}{4}\]
When we add like fractions, only the numerators will be added and the denominator remains the same.
Now, adding the terms, we get
\[ \Rightarrow {a^a} + {b^b} = \dfrac{{108 + 1}}{4}\]
Adding the numerators of the like fractions, we get
\[ \Rightarrow {a^a} + {b^b} = \dfrac{{109}}{4}\]
Therefore, \[{a^a} + {b^b} = \dfrac{{109}}{4}\]

Note: Here we can use cross multiplication method instead of L.C.M method. In the cross multiplication method, we cross multiply the numerator of the first fraction by the denominator of the second fraction. Then multiply the numerator of the second fraction by the denominator of the first fraction. Now, multiply both the denominators and take it as a common denominator. Then we can add the fractions now.