H Algebra 2/Trig ยท Unit 3
Multiply exponents, and take a root of a product one factor at a time.
A power of a power
Multiply the exponents.
\((a^{m})^{n} = a^{mn}\)
A product to a power
Raise the number and the variable separately.
\((ab)^{n} = a^{n}b^{n}\)
The answer is a whole-number power of x.
\((a^{m})^{n} = a^{mn}\)
\(x^{\frac{5}{3}\cdot 3}\)
\(x^{\frac{15}{3}}\)
\(x^{5}\)
Answer: \(x^{5}\)
\((a^{m})^{n} = a^{mn}\)
\(x^{\frac{2}{3}\cdot 3}\)
\(x^{\frac{6}{3}}\)
\(x^{2}\)
Answer: \(x^{2}\)
\((a^{m})^{n} = a^{mn}\)
\(x^{\frac{3}{4}\cdot 4}\)
\(x^{\frac{12}{4}}\)
\(x^{3}\)
Answer: \(x^{3}\)
\((a^{m})^{n} = a^{mn}\)
\(x^{\frac{4}{5}\cdot 5}\)
\(x^{\frac{20}{5}}\)
\(x^{4}\)
Answer: \(x^{4}\)
\((a^{m})^{n} = a^{mn}\)
\(x^{\frac{2}{5}\cdot 5}\)
\(x^{\frac{10}{5}}\)
\(x^{2}\)
Answer: \(x^{2}\)
\((a^{m})^{n} = a^{mn}\)
\(x^{\frac{5}{2}\cdot 2}\)
\(x^{\frac{10}{2}}\)
\(x^{5}\)
Answer: \(x^{5}\)
A whole number stays in front of x.
\((ab)^{n} = a^{n}b^{n}\)
\(\left(8x^{6}\right)^{\frac{1}{3}}\)
\(8^{\frac{1}{3}} \cdot \left(x^{6}\right)^{\frac{1}{3}}\)
\(2^{3} = 8\)
\(8^{\frac{1}{3}} = 2\)
\(\left(x^{6}\right)^{\frac{1}{3}} = x^{\frac{6}{3}}\)
\(x^{\frac{6}{3}} = x^{2}\)
\(2x^{2}\)
Answer: \(2x^{2}\)
\((ab)^{n} = a^{n}b^{n}\)
\(\left(27x^{3}\right)^{\frac{2}{3}}\)
\(27^{\frac{2}{3}} \cdot \left(x^{3}\right)^{\frac{2}{3}}\)
\(9^{3} = 27^{2}\)
\(27^{\frac{2}{3}} = 9\)
\(\left(x^{3}\right)^{\frac{2}{3}} = x^{\frac{6}{3}}\)
\(x^{\frac{6}{3}} = x^{2}\)
\(9x^{2}\)
Answer: \(9x^{2}\)
\((ab)^{n} = a^{n}b^{n}\)
\(\left(16x^{8}\right)^{\frac{3}{4}}\)
\(16^{\frac{3}{4}} \cdot \left(x^{8}\right)^{\frac{3}{4}}\)
\(8^{4} = 16^{3}\)
\(16^{\frac{3}{4}} = 8\)
\(\left(x^{8}\right)^{\frac{3}{4}} = x^{\frac{24}{4}}\)
\(x^{\frac{24}{4}} = x^{6}\)
\(8x^{6}\)
Answer: \(8x^{6}\)
\((ab)^{n} = a^{n}b^{n}\)
\(\left(32x^{10}\right)^{\frac{2}{5}}\)
\(32^{\frac{2}{5}} \cdot \left(x^{10}\right)^{\frac{2}{5}}\)
\(4^{5} = 32^{2}\)
\(32^{\frac{2}{5}} = 4\)
\(\left(x^{10}\right)^{\frac{2}{5}} = x^{\frac{20}{5}}\)
\(x^{\frac{20}{5}} = x^{4}\)
\(4x^{4}\)
Answer: \(4x^{4}\)
\((ab)^{n} = a^{n}b^{n}\)
\(\left(81x^{12}\right)^{\frac{1}{4}}\)
\(81^{\frac{1}{4}} \cdot \left(x^{12}\right)^{\frac{1}{4}}\)
\(3^{4} = 81\)
\(81^{\frac{1}{4}} = 3\)
\(\left(x^{12}\right)^{\frac{1}{4}} = x^{\frac{12}{4}}\)
\(x^{\frac{12}{4}} = x^{3}\)
\(3x^{3}\)
Answer: \(3x^{3}\)
\((ab)^{n} = a^{n}b^{n}\)
\(\left(125x^{3}\right)^{\frac{2}{3}}\)
\(125^{\frac{2}{3}} \cdot \left(x^{3}\right)^{\frac{2}{3}}\)
\(25^{3} = 125^{2}\)
\(125^{\frac{2}{3}} = 25\)
\(\left(x^{3}\right)^{\frac{2}{3}} = x^{\frac{6}{3}}\)
\(x^{\frac{6}{3}} = x^{2}\)
\(25x^{2}\)
Answer: \(25x^{2}\)
The exponent in the answer is still a fraction.
\((a^{m})^{n} = a^{mn}\)
\(x^{\frac{3}{4}\cdot 2}\)
\(x^{\frac{6}{4}}\)
\(x^{\frac{3}{2}}\)
Answer: \(x^{\frac{3}{2}}\)
\((a^{m})^{n} = a^{mn}\)
\(x^{\frac{5}{6}\cdot 2}\)
\(x^{\frac{10}{6}}\)
\(x^{\frac{5}{3}}\)
Answer: \(x^{\frac{5}{3}}\)
\((a^{m})^{n} = a^{mn}\)
\(x^{\frac{2}{5}\cdot 3}\)
\(x^{\frac{6}{5}}\)
Answer: \(x^{\frac{6}{5}}\)
\((ab)^{n} = a^{n}b^{n}\)
\(\left(27x^{4}\right)^{\frac{1}{3}}\)
\(27^{\frac{1}{3}} \cdot \left(x^{4}\right)^{\frac{1}{3}}\)
\(3^{3} = 27\)
\(27^{\frac{1}{3}} = 3\)
\(\left(x^{4}\right)^{\frac{1}{3}} = x^{\frac{4}{3}}\)
\(3x^{\frac{4}{3}}\)
Answer: \(3x^{\frac{4}{3}}\)
\((ab)^{n} = a^{n}b^{n}\)
\(\left(16x^{6}\right)^{\frac{1}{4}}\)
\(16^{\frac{1}{4}} \cdot \left(x^{6}\right)^{\frac{1}{4}}\)
\(2^{4} = 16\)
\(16^{\frac{1}{4}} = 2\)
\(\left(x^{6}\right)^{\frac{1}{4}} = x^{\frac{6}{4}}\)
\(x^{\frac{3}{2}}\)
\(2x^{\frac{3}{2}}\)
Answer: \(2x^{\frac{3}{2}}\)
\(x^{m} \cdot x^{n} = x^{m+n}\)
\(x^{\frac{1}{2}} \cdot x^{\frac{1}{3}}\)
\(x^{\frac{1}{2}+\frac{1}{3}}\)
\(x^{\frac{3}{6}+\frac{2}{6}}\)
\(x^{\frac{5}{6}}\)
Answer: \(x^{\frac{5}{6}}\)
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