When we have an expression raised to a power that, in turn, is raised (within parentheses) to another power, we can multiply the exponents and raise the base number to the result of this multiplication.
(43)2= We can see that the exponent 2 applies to the entire expression 43. therefore, we can multiply both exponents and raise the base to the result of the multiplication. We will obtain: 43Γ2=46=4096
If we were presented with an exercise in which there is a certain power over a term that already has another power, we will multiply the powers that have equal bases.
Example 2
Let's start with an easy one:
(X6β3)4=
We'll see that there is a subtraction in the exponents of X and that, first, we must deal with it.
We'll do this and obtain:
(X3)4=
Now we can apply the power of a power property and multiply the exponents, we will obtain:
X12
Good. Let's move on to a more complicated example:
Example 3
(X2X2β)4β (Y4Y2β)3=
Recommendation:
Before applying the power located outside the parentheses to each of the terms separately, first, it is advisable to carefully observe the exercise.
Upon observing it, you will realize that you can reduce or subtract exponents from the fractions themselves, before touching the exponent located outside the parentheses.
We will subtract the exponents of the corresponding bases (we will reduce) and obtain:
(2X)4β (4Y)3=
Now we can apply the exponent to each of the terms separately (do not forget about the coefficients) and we will get:
16X4β 64Y3=
We can try to find a common term to better organize the exercise and we will obtain:
16(X4β 4Y3)
Perfect! Now, let's move on to a complex and slightly different example:
Power of a power advanced examples:
Example 4
(2X+3)Xβ (2X)4=
Don't worry, even if there are mathematical operations among the exponents, the properties do not change.
Let's start with the first expression which is a bit more complex. We learned that, when we have a power of a power we multiply the exponents.
We will multiply the entire exponent that is inside the parentheses by the entire exponent located outside the parentheses. We will do the same with the other term and we will obtain:
2(X+3)β Xβ 24X=
We will multiply the exponents of the first expression and we will obtain:
2X2+3Xβ 24X=
Now let's remember that, if we have a multiplication operation between equal bases we can add the exponents.
We will do this and we will obtain:
2X2+3X+4X=
We simplify terms in the exponent and it will give us:
2X2+7X=
Example 5
Simplify the following expression:
(2xy2)3(3x3y2)2(2x2y4)4β
To simplify the expression, first apply the power of a product property, which allows us to raise each of the factors inside the parenthesis to the indicated power, then apply the power of a power property. We obtain: