MHB Simplifying expression with exponents

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SUMMARY

The expression $(-4x^n)$ simplifies to $(-4)^n \cdot x^n$, not $(-4)^n \cdot (-x)^n$. The negative sign applies only to the coefficient, which means $-4x^n$ is equivalent to $-4 \cdot x^n$. If the expression were $(-4x)^n$, it would equal $(-4)^n \cdot x^n$ or $4^n \cdot (-x)^n$. This distinction is crucial for correctly interpreting and simplifying expressions involving negative coefficients and variables.

PREREQUISITES
  • Understanding of algebraic expressions and exponents
  • Familiarity with the properties of negative numbers
  • Knowledge of the distributive property in multiplication
  • Basic skills in simplifying mathematical expressions
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  • Study the rules of exponents in algebra
  • Learn about the distributive property and its applications
  • Explore the implications of negative coefficients in expressions
  • Practice simplifying complex algebraic expressions with negative terms
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Students, educators, and anyone looking to deepen their understanding of algebraic expressions, particularly those involving exponents and negative coefficients.

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Supposing I have

$$(-4x^n)$$

Why does it equal $(-4)^n * x^n$ and not $(-4)^n * (-x) ^ n$?

When we have a negative symbol it only applies to the first item in an expression? so $-xbcw$ equals $b * c * w * (-x)$? Which means if we wanted the other items to be negative we would have to do $x(-b)(-c)w$ (for $b$ and $c$ to be negative and the other items to be positive)?
 
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Essentially you have $-4x^n$ which is equal to $-4\cdot x^n$.

If you intended $(-4x)^n$ then we have $(-4x)^n=(-4)^n\cdot x^n$ or $4^n\cdot (-x)^n$.

$-4\cdot-x=4x$ not $-4x$.

Does that help?
 
greg1313 said:
Essentially you have $-4x^n$ which is equal to $-4\cdot x^n$.

If you intended $(-4x)^n$ then we have $(-4x)^n=(-4)^n\cdot x^n$ or $4^n\cdot (-x)^n$.

$-4\cdot-x=4x$ not $-4x$.

Does that help?

So, in no uncertain terms, $(-4)^n\cdot x^n$ equals $4^n\cdot (-x)^n$ ?
 
Yes.
 

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