Answer:
a. [tex]-2x\sqrt[3]{x}[/tex]
b. [tex]\frac{1}{2^x}[/tex]
Step-by-step explanation:
a.
Original equation:
[tex]\frac{(-x)^4}{4x}*\frac{8(-x)^{-3}}{x^{-\frac{4}{3}}}[/tex]
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:
[tex]\frac{x^4}{4x}*\frac{8(-x)^{-3}}{x^{-\frac{4}{3}}}[/tex]
Cancel out one of the x's on the left fraction:
[tex]\frac{x^3}{4}*\frac{8(-x)^{-3}}{x^{-\frac{4}{3}}}[/tex]
Rewrite the exponent in the numerator: [tex]a^{-x} = \frac{1}{a^x}[/tex]
[tex]\frac{x^3}{4}*\frac{8*\frac{1}{-x^3}}{x^{-\frac{4}{3}}}[/tex]
Simplify the numerator:
[tex]\frac{x^3}{4}*\frac{\frac{8}{-x^3}}{x^{-\frac{4}{3}}}[/tex]
Keep numerator, change division to multiplication, flip the denominator:
[tex]\frac{x^3}{4}*\frac{8}{-x^3} * \frac{1}{x^{-\frac{4}{3}}}[/tex]
multiply the denominator using the exponent identity: [tex]x^a*x^b=x^{a+b}[/tex]
[tex]\frac{x^3}{4}*\frac{8}{-x^{\frac{5}{3}}}[/tex]
Multiply the numerators and denominators:
[tex]\frac{8x^3}{-4x^{\frac{5}{3}}}[/tex]
Use the fact that: [tex]\frac{x^a}{x^b}=x^{a-b}[/tex] to divide the x^3 and x^(5/3) and divide the 4 by the -8
[tex]-2x^{\frac{4}{3}}[/tex]
Rewrite the exponent using the exponent identity: [tex]x^{\frac{a}{b}} = \sqrt[b]{x^a}=\sqrt[b]{x}^a[/tex]
[tex]-2\sqrt[3]{x^4}[/tex]
Rewrite as two radicals: [tex]\sqrt[n]{a} * \sqrt[n]{b} = \sqrt[n]{ab}[/tex]
[tex]-2\sqrt[3]{x^3} * \sqrt[3]{x}[/tex]
Simplify:
[tex]-2x\sqrt[3]{x}[/tex]
b.
[tex]2^{2x}\div4^{3x}*64^{\frac{x}{2}}[/tex]
Rewrite the 4 as 2^2
[tex]2^{2x}\div(2^2)^{3x}*64^{\frac{x}{2}}[/tex]
Use the exponent identity: [tex](x^a)^b=x^{ab}[/tex]
[tex]2^{2x}\div2^{6x}}*64^{\frac{x}{2}}[/tex]
Use the exponent identity: [tex]\frac{x^a}{x^b}=x^{a-b}[/tex]
[tex]2^{2x-6x} = 2^{-4x}[/tex]
Rewrite this part using the definition of a negative exponent: [tex](\frac{a}{b})^{-x} = \frac{b}{a^x}[/tex].
[tex]\frac{1}{2^{4x}} * 64^{\frac{x}{2}}[/tex]
Multiply:
[tex]\frac{64^{\frac{x}{2}}}{2^{4x}}[/tex]
rewrite 64 as 2^6
[tex]\frac{(2^6)^{\frac{x}{2}}}{2^{4x}}[/tex]
Use the identity: [tex](x^a)^b=x^{ab}[/tex]
[tex]\frac{2^{3x}}{2^{4x}}[/tex]
Use the identity: [tex]\frac{x^a}{x^b}=x^{a-b}[/tex]
[tex]2^{-x}[/tex]
rewrite using the definition of a negative exponent: [tex](\frac{a}{b})^{-x} = \frac{b}{a^x}[/tex]
[tex]\frac{1}{2^x}[/tex]
