Question

What is the value of the quantity negative one half cubed all raised to the power of negative 3

Answer 1

Answer: -512

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Step-by-step explanation:

Let's focus on the portion
`\left(-(1)/(2)\right)^3` for now.

The exponent 3 means we'll have 3 copies of the base multiplied out like so

`\left(-(1)/(2)\right)^3 = \left(-(1)/(2)\right)*\left(-(1)/(2)\right)*\left(-(1)/(2)\right) = -(1)/(8)`

Two of the negatives pair up and cancel out. We're left with a negative sign in the result. Also 2*2*2 = 8, which explains the denominator.

In short,

`\left(-(1)/(2)\right)^3 = -(1)/(8)`

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We're then going to raise that result to the power of -3

So,

`\bigg[ \left(-(1)/(2)\right)^3 \bigg]^(-3) = \left(-(1)/(8)\right)^(-3) \\\\= (-8)^3 \\\\=(-8)*(-8)*(-8) \\\\= -512`

The second step in that section above is possible because of this exponent rule:
`a^(-b) = (1)/(a^b)`

The negative exponent means "flip the base to make the exponent positive". Think of -8 as -8/1.