Sure, let's break down the problem step-by-step.
We need to find which expression is equivalent to [tex]\( 6^8 \cdot 6^3 \)[/tex].
### Step 1: Use the properties of exponents
One of the fundamental properties of exponents states that when you multiply two exponential expressions with the same base, you add their exponents:
[tex]\[ a^m \cdot a^n = a^{m+n} \][/tex]
In this case, our base is [tex]\( 6 \)[/tex], [tex]\( m = 8 \)[/tex], and [tex]\( n = 3 \)[/tex]. Therefore:
[tex]\[ 6^8 \cdot 6^3 = 6^{8+3} \][/tex]
### Step 2: Simplify the exponent sum
Add the exponents together:
[tex]\[ 8 + 3 = 11 \][/tex]
So, we get:
[tex]\[ 6^8 \cdot 6^3 = 6^{11} \][/tex]
### Step 3: Compare with given options
Now we match [tex]\( 6^{11} \)[/tex] with the options provided:
1. [tex]\(\frac{12^{11}}{2^{11}}\)[/tex]:
This can be simplified using the property [tex]\( \left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \)[/tex]:
[tex]\[ \frac{12^{11}}{2^{11}} = \left(\frac{12}{2}\right)^{11} = 6^{11} \][/tex]
2. [tex]\( 6^{24} \)[/tex]:
Clearly, [tex]\( 6^{24} \neq 6^{11} \)[/tex].
3. [tex]\(\frac{1}{6^{11}}\)[/tex]:
This is the reciprocal of [tex]\( 6^{11} \)[/tex], so it does not equal [tex]\( 6^{11} \)[/tex].
4. [tex]\(\left(6^5\right)^6\)[/tex]:
Using the property [tex]\( (a^m)^n = a^{m \cdot n} \)[/tex]:
[tex]\[ \left(6^5\right)^6 = 6^{5 \cdot 6} = 6^{30} \][/tex]
This also does not equal [tex]\( 6^{11} \)[/tex].
### Conclusion
The expression equivalent to [tex]\( 6^8 \cdot 6^3 \)[/tex] is [tex]\(\frac{12^{11}}{2^{11}}\)[/tex].
