Let's simplify each expression step-by-step using the properties of exponents:
### Expression 1: [tex]\(\left(-8 c^0\right)^5\)[/tex]
1. Understanding [tex]\(c^0\)[/tex]: By the property of exponents, any nonzero number raised to the power of 0 is 1. Therefore, [tex]\(c^0 = 1\)[/tex].
2. Simplify the Expression:
[tex]\[
\left(-8 c^0\right)^5 = \left(-8 \cdot 1\right)^5 = \left(-8\right)^5
\][/tex]
3. Calculate [tex]\(\left(-8\right)^5\)[/tex]:
[tex]\[
\left(-8\right)^5 = (-8) \cdot (-8) \cdot (-8) \cdot (-8) \cdot (-8) = -32768
\][/tex]
So, the simplified expression is:
[tex]\[
-32768
\][/tex]
### Expression 2: [tex]\((-9 x)^3 - 9 x^3\)[/tex]
1. Simplify [tex]\((-9x)^3\)[/tex]:
[tex]\[
(-9x)^3 = (-9)^3 \cdot x^3 = -729 \cdot x^3 = -729x^3
\][/tex]
2. Subtract [tex]\(9x^3\)[/tex] from [tex]\(-729x^3\)[/tex]:
[tex]\[
(-729x^3) - (9x^3) = -729x^3 - 9x^3
\][/tex]
3. Combine like terms:
[tex]\[
-729x^3 - 9x^3 = -738x^3
\][/tex]
So, the simplified expression is:
[tex]\[
-738x^3
\][/tex]
### Expression 3: [tex]\(5 d^0 + 6 d^0\)[/tex]
1. Understanding [tex]\(d^0\)[/tex]: By the property of exponents, any nonzero number raised to the power of 0 is 1. Therefore, [tex]\(d^0 = 1\)[/tex].
2. Simplify the Expression:
[tex]\[
5 d^0 + 6 d^0 = 5 \cdot 1 + 6 \cdot 1 = 5 + 6
\][/tex]
So, the simplified expression is:
[tex]\[
11
\][/tex]
### Expression 4: [tex]\(\left(\frac{2 x^2 + 7 x + 4}{9}\right)^0\)[/tex]
1. Understanding [tex]\(\left(\text{anything}\right)^0\)[/tex]: By the property of exponents, any nonzero expression raised to the power of 0 is 1.
2. Simplify the Expression:
[tex]\[
\left(\frac{2 x^2 + 7 x + 4}{9}\right)^0 = 1
\][/tex]
So, the simplified expression is:
[tex]\[
1
\][/tex]
### Summary Table:
[tex]\[
\begin{tabular}{|c|c|}
\hline \multicolumn{2}{|c|}{Properties of Exponents} \\
\hline Simplify the following expressions where variables are nonzero numbers \\
\hline Original Expression & Simplified Expression \\
\hline \(\left(-8 c^0\right)^5\) & \(-32768\) \\
\hline \((-9 x)^3 - 9 x^3\) & \(-738x^3\) \\
\hline \(5 d^0 + 6 d^0\) & \(11\) \\
\hline \(\left(\frac{2 x^2 + 7 x + 4}{9}\right)^0\) & \(1\) \\
\hline
\end{tabular}
\][/tex]
