why is 5^0=1
Not understanding it. also
can you explain when (-3)^n is positive and when it is negative.
To understand why 5^0 equals 1, we need to consider the general rule for exponentiation. When we raise a number to the power of 0, regardless of the base, the result is always 1. This is a convention in mathematics, and it is defined this way because it helps maintain consistency and apply other rules of exponents.
One way to understand this is by considering the pattern that emerges when we simplify expressions of the form 5^n, where n is a positive integer:
5^3 = 5 × 5 × 5 = 125
5^2 = 5 × 5 = 25
5^1 = 5
Notice that each time we decrease the power by 1, we divide by 5. Following this pattern, if we divide by 5 one more time, we end up with:
5^0 = 1
So, 5^0 equals 1 because it is following the pattern of dividing by the base when we decrease the exponent.
As for the expression (-3)^n, whether it is positive or negative depends on whether the exponent, n, is even or odd.
If n is an even number (i.e., divisible by 2), like 2, 4, 6, and so on, then (-3)^n will always be positive. For example:
(-3)^2 = (-3) × (-3) = 9
(-3)^4 = (-3) × (-3) × (-3) × (-3) = 81
On the other hand, if n is an odd number, such as 1, 3, 5, and so on, then (-3)^n will always be negative:
(-3)^1 = -3
(-3)^3 = -3 × (-3) × (-3) = -27
In summary, when the exponent, n, of (-3)^n is even, the result is always positive. When the exponent is odd, the result is always negative.