Question

Can you help me find out what c is?

Answer 1

Answer:  "  $z = 10^(^3^m^-^2^n^)$  " .

________________________________

Step-by-step explanation:

________________________________

The question states:  "Can [you] help [me] find out what "c" is ? " .

________

       "Can [you] help [me] find out what "c" is ? " .

 →  We are to answer: "Part C" —within the "attached image" given:

________

 →   " $1000^m$ ÷ $100^n$ " ;  can be written in the form of:  " $10^z$ " .

 →     Express z  in terms of m and n .

________

Note: Rewrite:  " $1000^m$ " ;   and:  " $100^n$ " ;   as powers of "$10$" ;

  →  with "$10$" being the "base number" .

________

Let us start with:  " $1000^m$ " .

Note that the number:  " $1000 = 10^3$ " ;

     →   since " $1000$ " ;  has 3 (three) zeros after the "1" digit;

     → & since:  " $10^3 = 10 * 10* 10 = 1000$ " .

So:  →   $1000^m = (10^3)^m$ .

Now, let's simplify further.

Note the "multiplication property" of exponents:

    →  $(a^b)^c = a^(^b^*^c^)=a^(^b^c^)$ .

As such:  →   $1000^m=(10^3)^m = 10^(^3^*^m)=10^(^3^m^) .$

________

Likewise:  We can continue by rewriting our other term:

    →  " $100^n$ " ;  as a power of "$10$" ;

             with "$10$" being the "base number."

Note that the number:  " $100 = 10^2$ " ;

     →   since " $100$ " ;  has 2 (two) zeros after the "1" digit;

     →  & since:  " $10^2 = 10 * 10 = 100$ " .

So:

     →   $100^n = (10^2)^n$ .

Now, let's simplify further.

________

Again:  As aforementioned:

Note the "multiplication property" of exponents:

    →  $(a^b)^c = a^(^b^*^c^)=a^(^b^c^)$ .

As such:

     →  $100^n = (10^2)^n=10^(^2^*^n^) = 10^(^2^n^)$ .

________

Now, refer to the original problem:

     →  " $1000^m$  ÷  $100^n$  "  ;  

Rewrite this expression by substituting:

    →   " $10^(^3^m^)$ " ;  ←  [for:  " $1000^m$ ."] ;  And:

    →   " $10^(^2^n^)$ " ;   ←  [for:  " $100^n$ ."] ;

As follows:

    →  " $10^(^3^m^)$  ÷  $10^(^2^n^)$ "  .

________

Now:  Note the following "division property" of exponents:

________

    →  $\frac{a^m}{a^n} = a^(^m^-^n^)$ ;  $a\neq 0 .$

________

As such; let's rewrite our expression; and simplify—by plugging in our values into the "equation/formula" directly above; in which:

________

→  " $a$ " [in the aforementioned formula] is:   " $10$ " .

→  " $m$ " [the 'exponent' in the aforementioned formula] is:                                                                                                    

                                                                          " $(3m)$ " .

→  " $n$ " [the 'exponent' in the aforementioned formula is:

                                                                          " $(2n)$ " .

→  " $a^m$ " [in the aforementioned formula] is:  " $10^(^3^m^)$ " ;

→  " $a^n$ "  [in the aforementioned formula] is:  " $10^(^2^n^)$ " .

________

  →  Let's plug in our values;  and then "simplify" ; as follows:

________

→   $\frac{10^(^3^m^)}{10^(^2^n^)} =$ ? ;  

               $= 10^(^3^m^-^2^n^)$ .

→   $\frac{10^(^3^m^)}{10^(^2^n^)} = 10^(^3^m^-^2^n^)$    .

________

So:  →   " $10^z = ?$ " ;

    ↔   " $10^z = 10^(^3^m^-^2^n^)$ " .

         →  " $z = 10^(^3^m^-^2^n^)$ " .

_______________________________________

Hope this answer is helpful.

  Best wishes to you in your academic endeavors

          —and within the "Brainly" community!

_______________________________________