To evaluate [tex]\( 5^0 \)[/tex], we need to understand the properties of exponents.
For any non-zero number [tex]\( a \)[/tex], the expression [tex]\( a^0 \)[/tex] is defined to be 1. This is an important rule in exponentiation and can be understood through the following reasoning:
1. Pattern Recognition: Consider the pattern of decreasing exponents of a non-zero base. For instance:
[tex]\[
a^3 = a \cdot a \cdot a
\][/tex]
[tex]\[
a^2 = a \cdot a
\][/tex]
[tex]\[
a^1 = a
\][/tex]
Each time we decrease the exponent by 1, we are essentially dividing by the base [tex]\( a \)[/tex]:
[tex]\[
a^2 = a^3 / a
\][/tex]
[tex]\[
a^1 = a^2 / a
\][/tex]
Extending this pattern, when we decrease the exponent from 1 to 0, we divide once again by [tex]\( a \)[/tex]:
[tex]\[
a^0 = a^1 / a = a / a = 1
\][/tex]
2. Definition of Zero Exponent: By the definition of exponents, any non-zero base raised to the power of zero equals 1.
Therefore:
[tex]\[
5^0 = 1
\][/tex]
Conclusively, [tex]\( 5^0 = 1 \)[/tex].
