Answer:
[tex]\frac{12q^{\frac{7}{3}}}{p^{3}}[/tex]
Step-by-step explanation:
Here are some rules you need to simplify this expression:
Distribute exponents: When you raise an exponent to another exponent, you multiply the exponents together. This includes exponents that are fractions. [tex](a^{x})^{n} = a^{xn}[/tex]
Negative exponent rule: When an exponent is negative, you can make it positive by making the base a fraction. When the number is apart of a bigger fraction, you can move it to the other side (top/bottom). [tex]a^{-x} = \frac{1}{a^{x}}[/tex], and to help with this question: [tex]\frac{a^{-x}b}{1} = \frac{b}{a^{x}}[/tex].
Multiplying exponents with same base: When exponential numbers have the same base, you can combine them by adding their exponents together. [tex](a^{x})(a^{y}) = a^{x+y}[/tex]
Dividing exponents with same base: When exponential numbers have the same base, you can combine them by subtracting the exponents. [tex]\frac{a^{x}}{a^{y}} = a^{x-y}[/tex]
Fractional exponents as a radical: When a number has an exponent that is a fraction, the numerator can remain the exponent, and the denominator becomes the index (example, index here ∛ is 3). [tex]a^{\frac{m}{n}} = \sqrt[n]{a^{m}} = (\sqrt[n]{a})^{m}[/tex]
[tex]\frac{(8p^{-6} q^{3})^{2/3}}{(27p^{3}q)^{-1/3}}[/tex] Distribute exponent
[tex]=\frac{8^{(2/3)}p^{(-6*2/3)}q^{(3*2/3)}}{27^{(-1/3)}p^{(3*-1/3)}q^{(-1/3)}}[/tex] Simplify each exponent by multiplying
[tex]=\frac{8^{(2/3)}p^{(-4)}q^{(2)}}{27^{(-1/3)}p^{(-1)}q^{(-1/3)}}[/tex] Negative exponent rule
[tex]=\frac{8^{(2/3)}q^{(2)}27^{(1/3)}p^{(1)}q^{(1/3)}}{p^{(4)}}[/tex] Combine the like terms in the numerator with the base "q"
[tex]=\frac{8^{(2/3)}27^{(1/3)}p^{(1)}q^{(2)}q^{(1/3)}}{p^{(4)}}[/tex] Rearranged for you to see the like terms
[tex]=\frac{8^{(2/3)}27^{(1/3)}p^{(1)}q^{(2)+(1/3)}}{p^{(4)}}[/tex] Multiplying exponents with same base
[tex]=\frac{8^{(2/3)}27^{(1/3)}p^{(1)}q^{(7/3)}}{p^{(4)}}[/tex] 2 + 1/3 = 7/3
[tex]=\frac{\sqrt[3]{8^{2}}\sqrt[3]{27}p\sqrt[3]{q^{7}}}{p^{4}}[/tex] Fractional exponents as radical form
[tex]=\frac{(\sqrt[3]{64})(3)(p)(q^{\frac{7}{3}})}{p^{4}}[/tex] Simplified cubes. Wrote brackets to lessen confusion. Notice the radical of a variable can't be simplified.
[tex]=\frac{(4)(3)(p)(q^{\frac{7}{3}})}{p^{4}}[/tex] Multiply 4 and 3
[tex]=\frac{12pq^{\frac{7}{3}}}{p^{4}}[/tex] Dividing exponents with same base
[tex]=12p^{(1-4)}q^{\frac{7}{3}}[/tex] Subtract the exponent of 'p'
[tex]=12p^{(-3)}q^{\frac{7}{3}}[/tex] Negative exponent rule
[tex]=\frac{12q^{\frac{7}{3}}}{p^{3}}[/tex] Final answer
Here is a version in pen if the steps are hard to see.
