Absolutely, let's solve the given expression step by step.
The expression to simplify is: [tex]\(\frac{6 a^9 + 18 a^9}{-\left(-2 a^2\right)^4}\)[/tex].
Step 1: Combine terms in the numerator
Both terms in the numerator contain [tex]\(a^9\)[/tex] and can be combined:
[tex]\[6 a^9 + 18 a^9 = (6 + 18) a^9 = 24 a^9\][/tex]
Hence, the expression becomes:
[tex]\[\frac{24 a^9}{-\left(-2 a^2\right)^4}\][/tex]
Step 2: Simplify the denominator
The denominator is [tex]\(-\left(-2 a^2\right)^4\)[/tex].
First, simplify the inner part [tex]\((-2 a^2)^4\)[/tex]:
[tex]\[
(-2 a^2)^4 = (-2)^4 \cdot (a^2)^4 = 16 a^8
\][/tex]
Now, we account for the negative sign outside the parenthesis:
[tex]\[
- (16 a^8) = -16 a^8
\][/tex]
So the expression is now:
[tex]\[\frac{24 a^9}{-16 a^8}\][/tex]
Step 3: Simplify the fraction
[tex]\[
\frac{24 a^9}{-16 a^8} = \left(\frac{24}{-16}\right) \cdot \left(\frac{a^9}{a^8}\right)
\][/tex]
First, simplify the numerical part:
[tex]\[
\frac{24}{-16} = -\frac{24}{16} = -\frac{3}{2}
\][/tex]
Next, simplify the variable part using the rule of exponents [tex]\(a^m / a^n = a^{m-n}\)[/tex]:
[tex]\[
\frac{a^9}{a^8} = a^{9-8} = a^1 = a
\][/tex]
Putting it all together:
[tex]\[
-\frac{3}{2} \cdot a
\][/tex]
So, the simplified result is:
[tex]\[
-\frac{3}{2} a \quad \text{or} \quad -1.5 a
\][/tex]
Thus, the final simplified expression is:
[tex]\[
-\frac{3}{2} a
\][/tex]
