Answer :

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]