EXAMPLES:
Question:
Write
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$\,-(3x)^2\,$ in the form $\,kx^n\,$.
Solution:
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$-(3x)^2 = (-1)3^2x^2 = -9x^2\,$
or
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$-(3x)^2 = (-1)(3x)(3x) = (-1)(3\cdot 3)(x\cdot x) = -9x^2$
Question:
Write
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$\,-(2x)^3\,$ in the form $\,kx^n\,$.
Solution:
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$-(2x)^3 = (-1)2^3x^3 = -8x^3\,$
or
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$-(2x)^3 = (-1)(2x)(2x)(2x) = (-1)(2\cdot 2\cdot 2)(x\cdot x\cdot x) = -8x^3$
Question:
Write
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$\,-(-3x)^2\,$ in the form $\,kx^n\,$.
Solution:
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$-(-3x)^2 = (-1)(-3)^2x^2 = -9x^2\,$
or
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$-(-3x)^2 = (-1)(-3x)(-3x) = (-1)(-3\cdot -3)(x\cdot x) = -9x^2$
For mental math, the following thought process can be used:
- How many factors of $-1$ are there? Three (one outside, two inside); this is an odd number, so the answer is negative
- What's the size of the answer? $3^2 = 9$
- What's the variable part? $x^2$
- put it together to get $\,-9x^2$
Question:
Write
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$\,-(-2x)^3\,$ in the form $\,kx^n\,$.
Solution:
[beautiful math coming... please be patient]
$-(-2x)^3 = (-1)(-2)^3x^3 = 8x^3\,$
or
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$-(-2x)^3 = (-1)(-2x)(-2x)(-2x) = (-1)(-2\cdot -2\cdot -2)(x\cdot x\cdot x) = 8x^3$
For mental math, the following thought process can be used:
- How many factors of $-1$ are there? Four (one outside, three inside); this is an even number, so the answer is positive
- What's the size of the answer? $2^3 = 8$
- What's the variable part? $x^3$
- put it together to get $\,8x^3$
Helpful facts to remember:
$2^5 = 32$
$3^4 = 81$
$3^5 = 243$
$4^3 = 64$
$5^3 = 125$