Another rule of exponents in algebra is the zero exponent property. In this example, we multiply two expressions, each with the same base, and each with an exponent. Once you've read through the explanation, try the practice problems.
Explanation
Practice #1
Practice #2
| \(\small\mathsf{ (a^{n})(a^{m}) = a^{n + m} }\) | Using the Product of Powers Property, multiplying two exponential terms with the same base allows us to add the exponents together. |
| \(\small\mathsf{ (3^{2})(3^{?}) = 3^{2} }\) \(\small\mathsf{ 3^{2 + ?} = 3^{2} }\) \(\small\mathsf{ (3^{2})(3^{0}) = 3^{2} }\) |
Applying the product of powers property to the equation \(\small\mathsf{ (3^{2})(3^{?}) = 3^{2} }\), our equation can be rewritten where we add our exponents (the bases are the same, so the property applies). Adding our exponents 2 + 0 yields 2. |
| \(\small\mathsf{ (9)(3^{0}) = 9 }\) \(\small\mathsf{ 3^{0} = 1 }\) |
Since \(\small\mathsf{ (3^{2}) = 9 }\), we replace each \(\small\mathsf{ 3^{2} }\) with 9. Divide both sides by 9 and they cancel, leaving a 1. We are left with \(\small\mathsf{ 3^{0} = 1 }\). |
What is \(\mathsf{ 18^{0} ? }\)
\(\mathsf{ 18^{0} = 1 }\)
Any number raised to zero equals 1.
What is \(\mathsf{ (15x^{23}8^{19})^{0} = ? }\)
\(\mathsf{ (15x^{23}8^{19})^{0} = 1 }\)
Normally you would simplify terms inside the parenthesis, but if there is no addition or subtraction inside the parenthesis, and there is a zero exponent outside the parenthesis, then the answer is 1.
The zero exponent property allows us to set all powers of zero equal to one.
The Zero Exponent Property
\(\mathsf{ a^{0} = 1 }\)
Any base (except zero) raised to the zero power equals 1.
\(\mathsf{ 5^{0} = 1 }\)
\(\mathsf{ 223^{0} = 1 }\)
\(\mathsf{ (x^{3}y^{32})^{0} = 1 }\)
