High School

Expand and simplify the expressions in the table below:

[tex]
\[
\begin{array}{|l|l|l|}
\hline
\text{Original} & \text{Expand it!} & \text{Simplified Expression} \\
\hline
x^4 \cdot x^2 & x, x, x, x, x, x & x^6 \\
\hline
\left(m^2 p\right)\left(m^2 p^4\right) & m \cdot m \cdot p \cdot m \cdot m \cdot p \cdot p \cdot p \cdot p & m^4 p^5 \\
\hline
2 x^3 \cdot 7 x \cdot 5 x^3 & 2 \cdot 7 \cdot 5 \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x & 70 x^7 \\
\hline
\left(6 a^2 b\right)\left(4 a^3 b\right) & 6 \cdot 4 \cdot a \cdot a \cdot a \cdot a \cdot a \cdot b \cdot b & 24 a^5 b^2 \\
\hline
\left(-a^4\right)\left(-3 a^3\right) & -1 \cdot -3 \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a \cdot a & 3 a^7 \\
\hline
\end{array}
\]
[/tex]

Simplify the following expressions:

i) [tex]x^{42} \cdot x^{17} = x^{59}[/tex]

ii) [tex]x^9 \cdot x^7 \cdot x^4 \cdot x^{11} = x^{31}[/tex]

iii) [tex]\left(7 x^4\right) \cdot\left(-2 x^8\right) = -14 x^{12}[/tex]

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]