Answer :
We will simplify each given expression using the exponent rule that when multiplying powers with the same base you add their exponents. Detailed steps are provided below:
--------------------------------------------------------
1) Simplify
[tex]$$
x^{42} \cdot x^{17}.
$$[/tex]
Since the base is the same ([tex]$x$[/tex]), add the exponents:
[tex]$$
42+17=59.
$$[/tex]
So, the simplified result is:
[tex]$$
x^{59}.
$$[/tex]
--------------------------------------------------------
2) Simplify
[tex]$$
x^9 \cdot x^7 \cdot x^4 \cdot x^{11}.
$$[/tex]
Add all the exponents:
[tex]$$
9+7+4+11=31.
$$[/tex]
Thus, the expression simplifies to:
[tex]$$
x^{31}.
$$[/tex]
--------------------------------------------------------
3) Simplify
[tex]$$
(7x^4) \cdot (-2x^8).
$$[/tex]
First, multiply the coefficients:
[tex]$$
7 \times (-2) = -14.
$$[/tex]
Next, add the exponents for [tex]$x$[/tex]:
[tex]$$
4+8=12.
$$[/tex]
Thus, the expression simplifies to:
[tex]$$
-14x^{12}.
$$[/tex]
--------------------------------------------------------
In summary, the results are:
- [tex]$$x^{42}\cdot x^{17}=x^{59},$$[/tex]
- [tex]$$x^9\cdot x^7\cdot x^4\cdot x^{11}=x^{31},$$[/tex]
- [tex]$$(7x^4)\cdot(-2x^8)=-14x^{12}.$$[/tex]
--------------------------------------------------------
1) Simplify
[tex]$$
x^{42} \cdot x^{17}.
$$[/tex]
Since the base is the same ([tex]$x$[/tex]), add the exponents:
[tex]$$
42+17=59.
$$[/tex]
So, the simplified result is:
[tex]$$
x^{59}.
$$[/tex]
--------------------------------------------------------
2) Simplify
[tex]$$
x^9 \cdot x^7 \cdot x^4 \cdot x^{11}.
$$[/tex]
Add all the exponents:
[tex]$$
9+7+4+11=31.
$$[/tex]
Thus, the expression simplifies to:
[tex]$$
x^{31}.
$$[/tex]
--------------------------------------------------------
3) Simplify
[tex]$$
(7x^4) \cdot (-2x^8).
$$[/tex]
First, multiply the coefficients:
[tex]$$
7 \times (-2) = -14.
$$[/tex]
Next, add the exponents for [tex]$x$[/tex]:
[tex]$$
4+8=12.
$$[/tex]
Thus, the expression simplifies to:
[tex]$$
-14x^{12}.
$$[/tex]
--------------------------------------------------------
In summary, the results are:
- [tex]$$x^{42}\cdot x^{17}=x^{59},$$[/tex]
- [tex]$$x^9\cdot x^7\cdot x^4\cdot x^{11}=x^{31},$$[/tex]
- [tex]$$(7x^4)\cdot(-2x^8)=-14x^{12}.$$[/tex]