MHB Simplifying expression with exponents

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The expression $(-4x^n)$ simplifies to $(-4)^n \cdot x^n$ because the negative sign only applies to the coefficient, not the variable. In contrast, if the expression were $(-4x)^n$, it would equal $(-4)^n \cdot x^n$ or $4^n \cdot (-x)^n$. The discussion clarifies that $-4 \cdot -x$ results in $4x$, not $-4x$. Therefore, the correct interpretation hinges on the placement of parentheses and the application of the negative sign. Understanding these nuances is essential for accurate simplification of expressions involving exponents.
tmt1
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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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