Question - Simplifying Complex Expression with Fractions and Exponents
Solution:
To solve this expression, we should first simplify each part and then multiply them together. Let's start with the first part of the expression:The first part is a fraction with an exponent of -2:$$\left(\frac{{2p^3}}{{\sqrt[5]{q}}}\right)^{-2}$$Let's express the fifth root as fractional exponent and distribute the -2 exponent to each part inside the brackets:$$= \left(2p^3 \cdot q^{-\frac{1}{5}}\right)^{-2}$$$$= 2^{-2} \cdot \left(p^3\right)^{-2} \cdot \left(q^{-\frac{1}{5}}\right)^{-2}$$Now, apply the power of a power rule by multiplying the exponents:$$= 2^{-2} \cdot p^{-6} \cdot q^{\frac{2}{5}}$$The second part is:$$\frac{4}{p^{-1}q}$$Before multiplying both parts together, let's rewrite everything with positive exponents. For the first part, remember we need to move the terms with negative exponents to the denominator to make the exponents positive:$$= \frac{q^{\frac{2}{5}}}{2^2p^6}$$$$= \frac{q^{\frac{2}{5}}}{4p^6}$$For the second part:$$\frac{4}{p^{-1}q} = \frac{4p}{q}$$, because moving $$p^{-1}$$ to the numerator makes the exponent positive.Now, we multiply these two expressions together:$$\left(\frac{q^{\frac{2}{5}}}{4p^6}\right) \cdot \left(\frac{4p}{q}\right)$$Multiplying the numerators and the denominators separately:$$\frac{q^{\frac{2}{5}} \cdot 4p}{4p^6 \cdot q}$$Notice that $$4$$ in the numerator and denominator cancels out and one p from the numerator cancels with one p from the denominator:$$\frac{q^{\frac{2}{5}} \cdot p}{p^6 \cdot q}$$Split up the exponents for the q terms and subtract the exponents for the p terms:$$= \frac{q^{\frac{2}{5}}}{p^{6-1}q}$$$$= \frac{q^{\frac{2}{5}}}{p^5q}$$Now, subtract the exponents for q, since dividing powers with the same base means you subtract the exponents:$$= \frac{1}{p^5q^{1-\frac{2}{5}}}$$$$= \frac{1}{p^5q^{\frac{5}{5}-\frac{2}{5}}}$$$$= \frac{1}{p^5q^{\frac{3}{5}}}$$And that's our result with positive indices:\[ \frac{1}{p^5q^{\frac{3}{5}}} \]
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