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Answer :
We want to write each expression in simplest exponent form.
● For the expression
[tex]$$9 \cdot 9 \cdot 9 \cdot x \cdot x \cdot x \cdot x,$$[/tex]
notice that multiplying three 9’s is the same as raising 9 to the third power:
[tex]$$9 \cdot 9 \cdot 9 = 9^3.$$[/tex]
Since
[tex]$$9^3 = 729,$$[/tex]
and multiplying four instances of [tex]$x$[/tex] gives
[tex]$$x \cdot x \cdot x \cdot x = x^4,$$[/tex]
the simplified form is:
[tex]$$729 x^4.$$[/tex]
● For the expression
[tex]$$6 x^3,$$[/tex]
this is already written in simplest exponent form:
[tex]$$6x^3.$$[/tex]
● For the expression
[tex]$$9^3 x^4,$$[/tex]
the exponent form is already given. Since we computed that
[tex]$$9^3 = 729,$$[/tex]
the expression can also be written as:
[tex]$$729 x^4.$$[/tex]
● For the expression
[tex]$$6^5 x^3,$$[/tex]
first note that
[tex]$$6^5 = 7776,$$[/tex]
so the expression becomes:
[tex]$$7776 x^3.$$[/tex]
● For the expression
[tex]$$6 x^7,$$[/tex]
this expression is already in its simplest form:
[tex]$$6 x^7.$$[/tex]
Thus, the final simplified forms are:
[tex]$$
\begin{aligned}
9 \cdot 9 \cdot 9 \cdot x \cdot x \cdot x \cdot x &= 729 x^4,\\[1mm]
6x^3 &= 6x^3,\\[1mm]
9^3x^4 &= 729 x^4,\\[1mm]
6^5x^3 &= 7776 x^3,\\[1mm]
6x^7 &= 6 x^7.
\end{aligned}
$$[/tex]
● For the expression
[tex]$$9 \cdot 9 \cdot 9 \cdot x \cdot x \cdot x \cdot x,$$[/tex]
notice that multiplying three 9’s is the same as raising 9 to the third power:
[tex]$$9 \cdot 9 \cdot 9 = 9^3.$$[/tex]
Since
[tex]$$9^3 = 729,$$[/tex]
and multiplying four instances of [tex]$x$[/tex] gives
[tex]$$x \cdot x \cdot x \cdot x = x^4,$$[/tex]
the simplified form is:
[tex]$$729 x^4.$$[/tex]
● For the expression
[tex]$$6 x^3,$$[/tex]
this is already written in simplest exponent form:
[tex]$$6x^3.$$[/tex]
● For the expression
[tex]$$9^3 x^4,$$[/tex]
the exponent form is already given. Since we computed that
[tex]$$9^3 = 729,$$[/tex]
the expression can also be written as:
[tex]$$729 x^4.$$[/tex]
● For the expression
[tex]$$6^5 x^3,$$[/tex]
first note that
[tex]$$6^5 = 7776,$$[/tex]
so the expression becomes:
[tex]$$7776 x^3.$$[/tex]
● For the expression
[tex]$$6 x^7,$$[/tex]
this expression is already in its simplest form:
[tex]$$6 x^7.$$[/tex]
Thus, the final simplified forms are:
[tex]$$
\begin{aligned}
9 \cdot 9 \cdot 9 \cdot x \cdot x \cdot x \cdot x &= 729 x^4,\\[1mm]
6x^3 &= 6x^3,\\[1mm]
9^3x^4 &= 729 x^4,\\[1mm]
6^5x^3 &= 7776 x^3,\\[1mm]
6x^7 &= 6 x^7.
\end{aligned}
$$[/tex]
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