Question

Simplify the following:
A.
(-x)^4/4x * 8(-x)^-3/x^-3/4

Answer 1

Answer for (a) -2x^4/3

Answer for (b) 2^5x/4^3x

Answer 2

a.
`-2x\sqrt[3]{x}`

b.
`(1)/(2^x)`

Explanation:

a.

Original equation:

`((-x)^4)/(4x)*\frac{8(-x)^(-3)}{x^{-(4)/(3)}}`

So (-x)^4 can be seen as (-x * -x) * (-x * -x), which becomes x^2 * x^2 = x^4, the negatives cancel out of the degree is even. So it becomes:

`(x^4)/(4x)*\frac{8(-x)^(-3)}{x^{-(4)/(3)}}`

Cancel out one of the x's on the left fraction:

`(x^3)/(4)*\frac{8(-x)^(-3)}{x^{-(4)/(3)}}`

Rewrite the exponent in the numerator:
`a^(-x) = (1)/(a^x)`

`(x^3)/(4)*\frac{8*(1)/(-x^3)}{x^{-(4)/(3)}}`

Simplify the numerator:

`(x^3)/(4)*\frac{(8)/(-x^3)}{x^{-(4)/(3)}}`

Keep numerator, change division to multiplication, flip the denominator:

`(x^3)/(4)*(8)/(-x^3) * \frac{1}{x^{-(4)/(3)}}`

multiply the denominator using the exponent identity:
`x^a*x^b=x^(a+b)`

`(x^3)/(4)*\frac{8}{-x^{(5)/(3)}}`

Multiply the numerators and denominators:

`\frac{8x^3}{-4x^{(5)/(3)}}`

Use the fact that:
`(x^a)/(x^b)=x^(a-b)` to divide the x^3 and x^(5/3) and divide the 4 by the -8

`-2x^{(4)/(3)}`

Rewrite the exponent using the exponent identity:
`x^{(a)/(b)} = \sqrt[b]{x^a}=\sqrt[b]{x}^a`

`-2\sqrt[3]{x^4}`

Rewrite as two radicals:
`\sqrt[n]{a} * \sqrt[n]{b} = \sqrt[n]{ab}`

`-2\sqrt[3]{x^3} * \sqrt[3]{x}`

Simplify:

`-2x\sqrt[3]{x}`

b.

`2^(2x)/4^(3x)*64^{(x)/(2)}`

Rewrite the 4 as 2^2

`2^(2x)/(2^2)^(3x)*64^{(x)/(2)}`

Use the exponent identity:
`(x^a)^b=x^(ab)`

`2^(2x)/2^(6x)}*64^{(x)/(2)}`

Use the exponent identity:
`(x^a)/(x^b)=x^(a-b)`

`2^(2x-6x) = 2^(-4x)`

Rewrite this part using the definition of a negative exponent:
`((a)/(b))^(-x) = (b)/(a^x)`.

`(1)/(2^(4x)) * 64^{(x)/(2)}`

Multiply:

`\frac{64^{(x)/(2)}}{2^(4x)}`

rewrite 64 as 2^6

`\frac{(2^6)^{(x)/(2)}}{2^(4x)}`

Use the identity:
`(x^a)^b=x^(ab)`

`(2^(3x))/(2^(4x))`

Use the identity:
`(x^a)/(x^b)=x^(a-b)`

`2^(-x)`

rewrite using the definition of a negative exponent:
`((a)/(b))^(-x) = (b)/(a^x)`

`(1)/(2^x)`