We have seen various exponent laws for powers with the same base. CREATE and SOLVE a problem that says to 'SIMPLIFY' and that:
a) includes multiplying powers with the same base.
b) includes dividing powers with the same base .
c) includes a power of a power.
d) includes an negative exponent.
e) includes an exponent of 0.
f) includes a rational exponent.
g) combines three of the above exponent laws.
Solution
1)
2)
3)
"\\qquad\\qquad\n\\begin{aligned}\n&=\\bigg[(a+b)^8\\bigg]^{\\frac{1}{2}}\\\\\n&=(a+b)^{8\\times\\frac{1}{2}}\\\\\n&=(a+b)^4\n\\end{aligned}"
4)
"\\qquad\\qquad\n\\begin{aligned}\n&=\\frac{a^{-5}}{p^{-5}}\\\\\n&=\\frac{1}{a^5}\\times\\frac{p^5}{1}\\\\\n&=\\frac{p^5}{a^5}\\\\\n&=(\\frac{p}{a})^5\n\\end{aligned}"
5)
"\\qquad\\qquad\n\\begin{aligned}\n&=\\frac{(r+p)^5}{(r+p)^5}\\\\\n&=(r+p)^{5-5}\\\\\n&=(r+p)^0\\\\\n&=1\n\\end{aligned}"
f)
"\\qquad\\qquad\n\\begin{aligned}\n&=a^{(\\frac{2}{3}-\\frac{3}{4})}\\\\\n&=a^{-\\frac{1}{12}}\\\\\n&=\\frac{1}{a^\\frac{1}{12}}\n\\end{aligned}"
g)
"\\qquad\n\\begin{aligned}\n&=\\frac{a^8\\times b^{\\frac{3}{4}}}{a^3\\times b^3\\times c^{-6}}\\\\\n&=\\frac{a^{(8-5)}\\times b^{(\\frac{3}{4}-3)}}{c^{-6}}\\\\\n&=\\frac{a^3\\times b^{-\\frac{9}{4}}}{c^{-6}}\\\\\n&=\\frac{a^3\\times c^6}{b^{\\frac{9}{4}}}\\\\\n&=\\frac{a^3\\times (c^2)^3}{b^{\\frac{9}{4}}}\\\\\n&=\\frac{(ac^2)^3}{b^{\\frac{9}{4}}}\n\n\\end{aligned}"
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