Let's analyze the problem and find the value of [tex]\( a \)[/tex] for the monomial [tex]\( 3x^2 y^a z^a \)[/tex] given that its degree is 10.
First, recall that the degree of a monomial is the sum of the exponents of all variables present in the term.
For the monomial [tex]\( 3x^2 y^a z^a \)[/tex]:
- The exponent of [tex]\( x \)[/tex] is 2.
- The exponent of [tex]\( y \)[/tex] is [tex]\( a \)[/tex].
- The exponent of [tex]\( z \)[/tex] is also [tex]\( a \)[/tex].
Since we know that the degree of this monomial is 10, we can set up the following equation by summing the exponents:
[tex]\[ 2 + a + a = 10 \][/tex]
Simplify this equation:
[tex]\[ 2 + 2a = 10 \][/tex]
Next, solve for [tex]\( a \)[/tex]:
1. Subtract 2 from both sides of the equation:
[tex]\[ 2a = 8 \][/tex]
2. Divide both sides by 2:
[tex]\[ a = 4 \][/tex]
Therefore, the value of [tex]\( a \)[/tex] is:
[tex]\[ \boxed{4} \][/tex]
