Sure, let's simplify each expression step by step:
### 1. [tex]\((6x)^3\)[/tex]
First, let's expand the expression:
[tex]\[
(6x)^3 = 6^3 \cdot x^3
\][/tex]
Next, calculate [tex]\(6^3\)[/tex]:
[tex]\[
6^3 = 6 \cdot 6 \cdot 6 = 216
\][/tex]
So, the simplified expression is:
[tex]\[
(6x)^3 = 216x^3
\][/tex]
### 2. [tex]\((-2x)^2\)[/tex]
First, let's expand the expression:
[tex]\[
(-2x)^2 = (-2)^2 \cdot x^2
\][/tex]
Next, calculate [tex]\((-2)^2\)[/tex]:
[tex]\[
(-2)^2 = (-2) \cdot (-2) = 4
\][/tex]
So, the simplified expression is:
[tex]\[
(-2x)^2 = 4x^2
\][/tex]
### 3. [tex]\(3x^5 \cdot 2x^7 \cdot 7x^7\)[/tex]
First, let's group the constants and the variables together:
[tex]\[
3 \cdot 2 \cdot 7 \cdot x^5 \cdot x^7 \cdot x^7
\][/tex]
Next, simplify the constants:
[tex]\[
3 \cdot 2 \cdot 7 = 42
\][/tex]
Then, use the property of exponents ([tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]):
[tex]\[
x^5 \cdot x^7 \cdot x^7 = x^{5+7+7} = x^{19}
\][/tex]
So, the simplified expression is:
[tex]\[
3x^5 \cdot 2x^7 \cdot 7x^7 = 42x^{19}
\][/tex]
### 4. [tex]\(7(2x)^2\)[/tex]
First, let's expand the expression inside the parentheses:
[tex]\[
(2x)^2 = (2x) \cdot (2x) = 2^2 \cdot x^2
\][/tex]
Next, calculate [tex]\(2^2\)[/tex]:
[tex]\[
2^2 = 4
\][/tex]
So, the expression inside the parentheses becomes:
[tex]\[
(2x)^2 = 4x^2
\][/tex]
Now, multiply by 7:
[tex]\[
7 \cdot 4x^2 = 28x^2
\][/tex]
So, the simplified expression is:
[tex]\[
7(2x)^2 = 28x^2
\][/tex]
### Summary
The simplified expressions are as follows:
[tex]\[
\begin{array}{|l|l|}
\hline \text{Original Expression} & \text{Simplified Expression} \\
\hline (6x)^3 & 216x^3 \\
\hline (-2x)^2 & 4x^2 \\
\hline 3x^5 \cdot 2x^7 \cdot 7x^7 & 42x^{19} \\
\hline 7(2x)^2 & 28x^2 \\
\hline
\end{array}
\][/tex]
