College Algebra 7th Edition

Published by Brooks Cole
ISBN 10: 1305115546
ISBN 13: 978-1-30511-554-5

Chapter P, Prerequisites - Section P.4 - Rational Exponents and Radicals - P.4 Exercises: 82

Answer

$a^{\frac{3}{4}}$

Work Step by Step

RECALL: (i) $\sqrt[n]{a^m} = a^{\frac{m}{n}}$ (ii) $\dfrac{a^m}{a^n}=a^{m-n}, a\ne0$ (iii) $(am)^m = a^mb^m$ (iv) $(a^m)^n=a^{mn}$ (v) $a^0=1, a\ne0$ Use rule (i) above to obtain: $=\dfrac{(a^3b)^{\frac{1}{2}}}{(a^3b^2)^{\frac{1}{4}}}$ Use rule (iii) above to obtain: $=\dfrac{(a^3)^{\frac{1}{2}}b^{\frac{1}{2}}}{(a^3)^{\frac{1}{4}}(b^2)^{\frac{1}{4}}}$ Use rule (iv) above to obtain: $=\dfrac{a^{3\cdot\frac{1}{2}}b^{\frac{1}{2}}}{a^{3\cdot\frac{1}{4}}b^{2\cdot\frac{1}{4}}} \\=\dfrac{a^{\frac{3}{2}}b^{\frac{1}{2}}}{b^{\frac{3}{4}}b^{\frac{2}{4}}} \\=\dfrac{a^{\frac{3}{2}}b^{\frac{1}{2}}}{b^{\frac{3}{4}}b^{\frac{1}{2}}} $ Use rule (ii) above to obtain: $=a^{(\frac{3}{2}-\frac{3}{4})}b^{(\frac{1}{2}-\frac{1}{2})} \\=a^{(\frac{6}{4}-\frac{3}{4})}b^{0} \\=a^{\frac{3}{4}}b^0$ Use rule (v) above to obtain: $\\=a^{\frac{3}{4}}\cdot 1 \\=a^{\frac{3}{4}}$
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