Certainly! Let's solve the expression [tex]\(5^8 \cdot 5^{-5}\)[/tex] step by step using the laws of exponents.
### Step 1: Law of Exponents (Multiplying Like Bases)
When you multiply two exponents with the same base, you add the exponents. The general rule is:
[tex]\[ a^m \cdot a^n = a^{m+n} \][/tex]
### Step 2: Applying the Law to the Given Exponents
Here, our base [tex]\(a\)[/tex] is 5, the exponents are 8 and -5. Using the law of exponents mentioned:
[tex]\[ 5^8 \cdot 5^{-5} = 5^{8 + (-5)} = 5^{8 - 5} \][/tex]
### Step 3: Simplifying the Exponent
[tex]\[ 8 - 5 = 3 \][/tex]
So,
[tex]\[ 5^8 \cdot 5^{-5} = 5^3 \][/tex]
### Step 4: Calculating the Result
Now we need to calculate [tex]\(5^3\)[/tex]:
[tex]\[ 5^3 = 5 \times 5 \times 5 = 125 \][/tex]
### Conclusion
Hence, the expression [tex]\(5^8 \cdot 5^{-5}\)[/tex] simplifies to [tex]\(5^3\)[/tex] and the value is [tex]\(125\)[/tex].
Therefore, the final answer is:
[tex]\[ 5^8 \cdot 5^{-5} = 5^3 = 125 \][/tex]
So, [tex]\( (5^8 \cdot 5^{-5}) = 125 \)[/tex].
