To solve for the value of [tex]\(n\)[/tex] that makes the statement true:
[tex]\[
6 x^n \cdot 4 x^2 = 24 x^6
\][/tex]
we will follow these steps:
1. Simplify the left-hand side of the equation:
[tex]\[
6 x^n \cdot 4 x^2
\][/tex]
First, we can multiply the numerical coefficients:
[tex]\[
6 \cdot 4 = 24
\][/tex]
Next, we need to combine the exponents of [tex]\(x\)[/tex] because when multiplying powers with the same base, we add the exponents:
[tex]\[
x^n \cdot x^2 = x^{n+2}
\][/tex]
Therefore, the left-hand side simplifies to:
[tex]\[
24 x^{n+2}
\][/tex]
2. Set the simplified left-hand side equal to the right-hand side of the equation:
[tex]\[
24 x^{n+2} = 24 x^6
\][/tex]
3. Divide both sides by 24 to isolate the terms with [tex]\(x\)[/tex]:
[tex]\[
x^{n+2} = x^6
\][/tex]
4. Since the bases (the x's) are the same, the exponents must be equal:
[tex]\[
n + 2 = 6
\][/tex]
5. Solve for [tex]\(n\)[/tex]:
[tex]\[
n + 2 = 6
\][/tex]
Subtract 2 from both sides:
[tex]\[
n = 6 - 2
\][/tex]
[tex]\[
n = 4
\][/tex]
Therefore, the value of [tex]\(n\)[/tex] that makes the statement true is:
[tex]\[
\boxed{4}
\][/tex]
