A b c whole cube

General
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a b c whole cube

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What does LectureNotes mean by ( (abc)^3 )?

Answer:

When referring to the expression ( (abc)^3 ), mathematically, this denotes the cube of the product of three variables or constants ( a ), ( b ), and ( c ). Instead of cubing each variable separately, the entire product is treated as a single term, which is then raised to the power of three.

Step 1: Understand the Expression

  • Expression: ( (abc)^3 )
  • Core Idea: This is an application of the exponentiation rule, which states that for any numbers or variables ( x, y, z ), the expression ( (xyz)^3 ) means ( (xyz) \times (xyz) \times (xyz) ).

Step 2: Simplifying the Expression

  1. Break it Down:

    • Begin with the product of the variables: ( abc ).
    • Multiply ( abc ) by itself three times:
      (abc) \times (abc) \times (abc)

  2. Using Exponent Rules:

    • The exponent rule for products tells us that ( (x \times y \times z)^n = x^n \times y^n \times z^n ).
    • Therefore, applying this rule here gives:
      (abc)^3 = a^3 \times b^3 \times c^3

Step 3: Conclusion

  • Interpretation: When raised to a power, each component of a multiplicative term shares this power.
  • Application: This principle is useful in algebra for simplifying expressions, solving equations, and performing polynomial multiplication.

Final Answer:

The expression ( (abc)^3 ) is equivalent to ( a^3 \times b^3 \times c^3 ). This application of the exponent rules highlights the importance of distributing the power over the entire product.

This explanation can be useful in understanding polynomial identities, transformations, and various algebraic manipulations. By recognizing that for any variables or constants within a multiplication, the cube affects each term equally, students gain a deeper understanding of algebraic operations.