Sure, let's solve the equation step-by-step.
We start with the given equation:
[tex]\[ 5^m \div 5^{-3} = 5 \][/tex]
Step 1: Recall the properties of exponents. Specifically, for any base [tex]\(a\)[/tex] and exponents [tex]\(m\)[/tex] and [tex]\(n\)[/tex],
[tex]\[ \frac{a^m}{a^n} = a^{m-n} \][/tex]
Applying this property to our equation, we get:
[tex]\[ 5^m \div 5^{-3} = 5^{m - (-3)} = 5^{m + 3} \][/tex]
Step 2: We are now left with
[tex]\[ 5^{m + 3} = 5 \][/tex]
Step 3: Recognize that [tex]\(5\)[/tex] can be written as [tex]\(5^1\)[/tex]. So, the equation becomes:
[tex]\[ 5^{m + 3} = 5^1 \][/tex]
Step 4: Since the bases are the same, we can set the exponents equal to each other:
[tex]\[ m + 3 = 1 \][/tex]
Step 5: Solve for [tex]\(m\)[/tex] by isolating the variable:
[tex]\[ m + 3 = 1 \][/tex]
[tex]\[ m = 1 - 3 \][/tex]
[tex]\[ m = -2 \][/tex]
Thus, the value of [tex]\(m\)[/tex] is [tex]\(-2\)[/tex].
