If the bases are the same, and they are nonzero, we can multiply powers by adding the exponents.
a^m * a^n = a^(m+n)
In this case we want to rewrite y^6 as the product of two same base powers y^n * y^m. According to the property for multiplying powers, and since the exponent is 6, we just need to chose two numbers n,m that add up to six. Remember that we are restricted to using positive numbers. We can find four examples in the table below.
y^n
y^m
y^n* y^m=y^(n+m)
y^1
y^5
y^1* y^5 =y^(1+5)=y^6
y^2
y^4
y^2* y^4 =y^(2+4)=y^6
y^3
y^3
y^3* y^3 =y^(3+3)=y^6
y^(12)
y^(112)
y^(12)* y^(112) =y^(12+ 112)=y^6
Notice that these are only some examples, as there are infinitely many solutions satisfying the required conditions.
b According to the property for multiplying powers, to rewrite y^6 as the product of two same base powers y^n * y^m, we just need to chose two numbers n and m that add up to six. Remember that at least one of them must be less than or equal to 0. We can find four examples in the table below.
y^n
y^m
y^n* y^m=y^(n+m)
y^0
y^6
y^0* y^6 =y^(0+6)=y^6
y^(-1)
y^7
y^(-1)* y^7 =y^(-1+7)=y^6
y^(-2)
y^8
y^(-2)* y^8 =y^(-2+8)=y^6
y^(-3)
y^9
y^(-3)* y^9 =y^(-3+9)=y^6
Notice that these are only some examples, as there are infinitely many solutions satisfying the required conditions.
c As we have seen in Part A and Part B, there are infinitely many numbers n and m that when added together get 6. Therefore, we can write infinitely many products of the form y^n* y^m = y^(n+m) which equals y^6.
