Identify the equivalent expression for each of the expressions below.
(m^1/3)(m^1/5)^0
The correct Answer and Explanation is:
To simplify the expression ((m^{1/3})(m^{1/5})^0), we need to apply the rules of exponents.
Step 1: Simplify ((m^{1/5})^0)
According to the exponent rule, any non-zero number raised to the power of zero is equal to 1. Therefore, we have:
[
(m^{1/5})^0 = 1
]
Step 2: Substitute back into the expression
Now, substituting this result back into the original expression gives us:
[
(m^{1/3})(m^{1/5})^0 = (m^{1/3})(1)
]
This simplifies to:
[
m^{1/3}
]
Final Result
Thus, the equivalent expression for ((m^{1/3})(m^{1/5})^0) is simply:
[
\boxed{m^{1/3}}
]
Explanation of Exponent Rules
- Zero Exponent Rule: The fundamental rule applied here is that any non-zero base raised to the power of zero equals one. This rule is crucial in simplifying expressions where an exponent is zero, as it essentially nullifies the base’s contribution in multiplication.
- Multiplication of Exponents: When multiplying exponential expressions with the same base, the exponents are added. This rule is helpful when combining different powers of the same base, although in this case, it wasn’t applied directly because of the zero exponent.
- Maintaining Base Consistency: It’s important to keep the base the same (in this case, (m)) throughout the manipulation of the expression. This ensures that any changes we make conform to the rules of exponents and maintain the integrity of the expression.
In conclusion, understanding these exponent rules allows us to simplify complex expressions effectively. In this specific case, we were able to reduce the expression involving multiple exponents down to a simple form using these foundational principles. The final equivalent expression is (m^{1/3}).