To simplify the expression [tex]\(\frac{2^7 \cdot 6^8 \cdot 8^{10}}{2 \cdot 6^6 \cdot 8^8}\)[/tex], we will break down the simplification step-by-step for each component of the numerator and the denominator.
### Step 1: Simplifying the powers of 2
- The numerator has [tex]\(2^7\)[/tex].
- The denominator has [tex]\(2\)[/tex], which is [tex]\(2^1\)[/tex].
We can subtract the exponents of 2:
[tex]\[
\frac{2^7}{2^1} = 2^{7 - 1} = 2^6
\][/tex]
### Step 2: Simplifying the powers of 6
- The numerator has [tex]\(6^8\)[/tex].
- The denominator has [tex]\(6^6\)[/tex].
We can subtract the exponents of 6:
[tex]\[
\frac{6^8}{6^6} = 6^{8 - 6} = 6^2
\][/tex]
### Step 3: Simplifying the powers of 8
- The numerator has [tex]\(8^{10}\)[/tex].
- The denominator has [tex]\(8^8\)[/tex].
We can subtract the exponents of 8:
[tex]\[
\frac{8^{10}}{8^8} = 8^{10 - 8} = 8^2
\][/tex]
### Step 4: Combining the simplified components
Now we have:
[tex]\[
2^6 \cdot 6^2 \cdot 8^2
\][/tex]
We will now write the values of these:
- [tex]\(2^6 = 64\)[/tex]
- [tex]\(6^2 = 36\)[/tex]
- [tex]\(8^2 = 64\)[/tex]
### Step 5: Computing the final result
We now multiply these simplified values together:
[tex]\[
64 \cdot 36 \cdot 64
\][/tex]
The computed result is:
[tex]\[
64 \cdot 36 \cdot 64 = 147456
\][/tex]
Therefore, the simplified expression [tex]\(\frac{2^7 \cdot 6^8 \cdot 8^{10}}{2 \cdot 6^6 \cdot 8^8}\)[/tex] equals 147,456.
