Certainly! Let's break down the expression [tex]\((-3x)^4\)[/tex] step by step to write it in expanded form:
1. Identify the components of the expression:
- The base inside the parentheses is [tex]\(-3x\)[/tex].
- The exponent outside the parentheses is [tex]\(4\)[/tex].
2. Separate the base into its individual factors:
- The base [tex]\(-3x\)[/tex] can be viewed as two separate factors: [tex]\(-3\)[/tex] and [tex]\(x\)[/tex].
3. Apply the exponent to each factor separately:
- According to the rules of exponents, when a product is raised to an exponent, each factor in the product is raised to that exponent. Mathematically, [tex]\((ab)^n = a^n \cdot b^n\)[/tex].
- Here, [tex]\(a = -3\)[/tex] and [tex]\(b = x\)[/tex], and [tex]\(n = 4\)[/tex].
4. Raise [tex]\(-3\)[/tex] to the power of 4:
- [tex]\((-3)^4\)[/tex] means multiplying [tex]\(-3\)[/tex] by itself four times: [tex]\(-3 \cdot -3 \cdot -3 \cdot -3\)[/tex].
- The result of this multiplication is [tex]\(81\)[/tex], as determined previously.
5. Raise [tex]\(x\)[/tex] to the power of 4:
- [tex]\(x^4\)[/tex] remains as it is in symbolic form since we are not multiplying numerical values for [tex]\(x\)[/tex].
6. Combine the results:
- Each factor raised to the 4th power is combined using multiplication: [tex]\(81\)[/tex] and [tex]\(x^4\)[/tex].
Therefore, the expanded form of [tex]\((-3x)^4\)[/tex] is:
[tex]\[ (81) \cdot (x^4) \][/tex]
So, the final expression is:
[tex]\[ (81) \cdot (x^4) \][/tex]
This concludes the step-by-step solution to expand the expression [tex]\((-3x)^4\)[/tex]. Remember to use parentheses to indicate multiplication when writing the final answer.
