Let's break down the steps to simplifying the product [tex]\((2x^3)^3\)[/tex] and provide the justifications for each step methodically:
Step 1: [tex]\((2 x^3)^3 = 2 x^3 \cdot 2 x^3 \cdot 2 x^3\)[/tex]
Justification: Apply the Power of a Product Property, which states that [tex]\((ab)^n = a^n \cdot b^n\)[/tex]. Here, we are essentially expressing the product as a repeated multiplication.
Step 2: [tex]\(2 x^3 \cdot 2 x^3 \cdot 2 x^3 = 2 \cdot 2 \cdot 2 \cdot x^3 \cdot x^3 \cdot x^3\)[/tex]
Justification: Simplify each term separately. We separate the coefficients and the variables to handle them independently.
Step 3: [tex]\(2 \cdot 2 \cdot 2 \cdot x^3 \cdot x^3 \cdot x^3 = (2 \cdot 2 \cdot 2) \cdot (x^3 \cdot x^3 \cdot x^3)\)[/tex]
Justification: Rearrange and group like terms. Group all coefficients together and all the variables together to prepare for further simplification.
Step 4: [tex]\((2 \cdot 2 \cdot 2) \cdot (x^3 \cdot x^3 \cdot x^3) = 8 x^9\)[/tex]
Justification: Multiply the exponents together. Simplify the numerical coefficients [tex]\(2 \cdot 2 \cdot 2 = 8\)[/tex] and apply the property of powers [tex]\((x^a \cdot x^b \cdot x^c = x^{a+b+c})\)[/tex] to get [tex]\(x^{3+3+3} = x^9\)[/tex].
Summarizing, the justifications for each step are:
- Apply the Power of a Product Property.
- Simplify each term separately.
- Rearrange and group like terms.
- Multiply the exponents together.
Through these steps and justifications, we have simplified [tex]\((2 x^3)^3\)[/tex] to [tex]\(8 x^9\)[/tex].
