To find the value of [tex]\( m \)[/tex] such that the equation [tex]\( 5^m \div 5^{-3} = 5^{-5} \)[/tex] holds true, follow these steps:
1. Understand the given equation:
[tex]\( 5^m \div 5^{-3} = 5^{-5} \)[/tex]
2. Recall the properties of exponents, particularly the division property:
[tex]\( \frac{a^m}{a^n} = a^{m-n} \)[/tex]
Applying this property to our equation, we have:
[tex]\( 5^m \div 5^{-3} = 5^{m - (-3)} \)[/tex]
Which simplifies to:
[tex]\( 5^{m + 3} \)[/tex]
3. Set the expressions equal to each other:
According to our equation,
[tex]\( 5^{m + 3} = 5^{-5} \)[/tex]
4. Since the bases are equal, we can equate the exponents:
[tex]\( m + 3 = -5 \)[/tex]
5. Solve for [tex]\( m \)[/tex]:
Subtract 3 from both sides to isolate [tex]\( m \)[/tex]:
[tex]\( m + 3 - 3 = -5 - 3 \)[/tex]
[tex]\( m = -8 \)[/tex]
So, the value of [tex]\( m \)[/tex] is:
[tex]\[
m = -8
\][/tex]
