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We need to develop the expression
[tex]\mleft(-\frac{1}{4}\mright)^3[/tex]
Step 1
Distribute the exponent to the numerator and to the denominator:
[tex]\mleft(-\frac{1}{4}\mright)^3=\frac{(-1)^{3}}{4^{3}}[/tex]
Step 2
Use the property
[tex]x^n=x\cdot x\cdot x\cdot\ldots\cdot x\text{ (n factors)}[/tex]
We obtain:
[tex]\begin{gathered} (-1)^3=(-1)\cdot(-1)\cdot(-1) \\ \\ 4^3=4\cdot4\cdot4 \end{gathered}[/tex]
Step 3
Use the properties:
• factors with the same sign result in a positive product
[tex](-1)\cdot(-1)=1[/tex]
• factors with opposite signs result in a negative product
[tex]1\cdot(-1)=-1[/tex]
Thus:
[tex](-1)\cdot(-1)\cdot(-1)=1\cdot(-1)=-1[/tex]
Step 4
Compute the product
[tex]4\cdot4\cdot4=16\cdot4=64[/tex]
Answer
Use the previous results to write:
[tex]\frac{(-1)^3}{4^3}=-\frac{1}{64}[/tex]
