To determine the value of [tex]\( m \)[/tex] in the given expression [tex]\( 4 x^5 y^m z \)[/tex], we need to understand the concept of the degree of a polynomial.
The degree of a polynomial is the sum of the exponents of the variables in each term, and for a single term polynomial, it is simply the sum of the exponents of the variables in that term.
Let's analyze the term [tex]\( 4 x^5 y^m z \)[/tex]:
- The exponent of [tex]\( x \)[/tex] is [tex]\( 5 \)[/tex].
- The exponent of [tex]\( y \)[/tex] is [tex]\( m \)[/tex].
- The exponent of [tex]\( z \)[/tex] is [tex]\( 1 \)[/tex].
The degree of the expression is given to be [tex]\( 10 \)[/tex]. Hence, we can write:
[tex]\[
\text{Degree of the expression} = \text{Exponent of } x + \text{Exponent of } y + \text{Exponent of } z
\][/tex]
[tex]\[
10 = 5 + m + 1
\][/tex]
Now, solve for [tex]\( m \)[/tex]:
[tex]\[
10 = 5 + m + 1
\][/tex]
[tex]\[
10 = 6 + m
\][/tex]
[tex]\[
m = 10 - 6
\][/tex]
[tex]\[
m = 4
\][/tex]
Hence, the value of [tex]\( m \)[/tex] is [tex]\( 4 \)[/tex].
