Simplify the expression $5^{3m+1} \div 5^{3m+3}$.
Understand the Problem
The question requires simplifying the expression $5^{3m+1} \div 5^{3m+3}$. This involves using the properties of exponents, specifically the rule for dividing exponents with the same base: $a^m / a^n = a^{(m-n)}$. We need to subtract the exponents and simplify to get the final result.
Answer
$\frac{1}{25}$
Answer for screen readers
$\frac{1}{25}$
Steps to Solve
- Apply the quotient rule of exponents
To divide exponential terms with the same base, we subtract the exponents:
$5^{3m+1} \div 5^{3m+3} = 5^{(3m+1)-(3m+3)}$
- Simplify the exponent
Distribute the negative sign and combine like terms:
$5^{(3m+1)-(3m+3)} = 5^{3m+1-3m-3} = 5^{-2}$
- Express the result with a positive exponent
Using the rule $a^{-n} = \frac{1}{a^n}$:
$5^{-2} = \frac{1}{5^2}$
- Calculate the final value
Evaluate $5^2$:
$\frac{1}{5^2} = \frac{1}{25}$
$\frac{1}{25}$
More Information
The expression simplifies to $\frac{1}{25}$, which is the result of applying exponent rules for division and negative exponents.
Tips
A common mistake is not distributing the negative sign correctly when subtracting the exponents, which can lead to an incorrect simplification of the exponent. Also, forgetting that $a^{-n} = \frac{1}{a^n}$ is a common mistake.
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