To determine which of these equations is justified by the commutative property of multiplication, let's review each option with a step-by-step approach. The commutative property of multiplication states that for any two numbers [tex]\(a\)[/tex] and [tex]\(b\)[/tex], the equation [tex]\(a \cdot b = b \cdot a\)[/tex] holds true.
Option A:
[tex]\[
4x \left(\frac{1}{4x}\right) = 1
\][/tex]
- Here we are looking at the multiplication of [tex]\(4x\)[/tex] and [tex]\(\frac{1}{4x}\)[/tex]. This equation is simplified using the property of multiplicative inverses, not the commutative property.
Option B:
[tex]\[
8(6x) = (8 \cdot 6)x
\][/tex]
- This transformation is distributing the multiplication inside the parentheses, which is an example of the associative property of multiplication rather than the commutative property.
Option C:
[tex]\[
2(x + 7) = 2x + 14
\][/tex]
- This equation uses the distributive property, where [tex]\(2\)[/tex] is multiplied across the terms inside the parentheses. This does not demonstrate the commutative property.
Option D:
[tex]\[
10(9x) = (9x) \cdot 10
\][/tex]
- This equation showcases the commutative property of multiplication, where the order of multiplication is switched. It shows that multiplying [tex]\(10\)[/tex] by [tex]\(9x\)[/tex] is the same as multiplying [tex]\(9x\)[/tex] by [tex]\(10\)[/tex].
Thus, the only equation that is justified by the commutative property of multiplication is [tex]\(10(9x) = (9x) \cdot 10\)[/tex], which corresponds to:
Option D.
