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.
\(\left(8x^{6}\right)^{\frac{1}{3}}\)
\(8^{\frac{1}{3}} \cdot \left(x^{6}\right)^{\frac{1}{3}}\)
\(8^{\frac{1}{3}} \cdot x^{\left(6\cdot \frac{1}{3}\right)}\)
\(8^{\frac{1}{3}} \cdot x^{2}\)
\(\sqrt[3]{8} \cdot x^{2}\)
\(2x^{2}\)
Answer: \(2x^{2}\)
\(\left(27x^{3}\right)^{\frac{2}{3}}\)
\(27^{\frac{2}{3}} \cdot \left(x^{3}\right)^{\frac{2}{3}}\)
\(27^{\frac{2}{3}} \cdot x^{\left(3\cdot \frac{2}{3}\right)}\)
\(27^{\frac{2}{3}} \cdot x^{2}\)
\(\left(\sqrt[3]{27}\right)^{2} \cdot x^{2}\)
\(3^{2} \cdot x^{2}\)
\(9x^{2}\)
Answer: \(9x^{2}\)
\(\left(16x^{8}\right)^{\frac{3}{4}}\)
\(16^{\frac{3}{4}} \cdot \left(x^{8}\right)^{\frac{3}{4}}\)
\(16^{\frac{3}{4}} \cdot x^{\left(8\cdot \frac{3}{4}\right)}\)
\(16^{\frac{3}{4}} \cdot x^{6}\)
\(\left(\sqrt[4]{16}\right)^{3} \cdot x^{6}\)
\(2^{3} \cdot x^{6}\)
\(8x^{6}\)
Answer: \(8x^{6}\)
\(\left(32x^{10}\right)^{\frac{2}{5}}\)
\(32^{\frac{2}{5}} \cdot \left(x^{10}\right)^{\frac{2}{5}}\)
\(32^{\frac{2}{5}} \cdot x^{\left(10\cdot \frac{2}{5}\right)}\)
\(32^{\frac{2}{5}} \cdot x^{4}\)
\(\left(\sqrt[5]{32}\right)^{2} \cdot x^{4}\)
\(2^{2} \cdot x^{4}\)
\(4x^{4}\)
Answer: \(4x^{4}\)
\(\left(81x^{12}\right)^{\frac{1}{4}}\)
\(81^{\frac{1}{4}} \cdot \left(x^{12}\right)^{\frac{1}{4}}\)
\(81^{\frac{1}{4}} \cdot x^{\left(12\cdot \frac{1}{4}\right)}\)
\(81^{\frac{1}{4}} \cdot x^{3}\)
\(\sqrt[4]{81} \cdot x^{3}\)
\(3x^{3}\)
Answer: \(3x^{3}\)
\(\left(125x^{3}\right)^{\frac{2}{3}}\)
\(125^{\frac{2}{3}} \cdot \left(x^{3}\right)^{\frac{2}{3}}\)
\(125^{\frac{2}{3}} \cdot x^{\left(3\cdot \frac{2}{3}\right)}\)
\(125^{\frac{2}{3}} \cdot x^{2}\)
\(\left(\sqrt[3]{125}\right)^{2} \cdot x^{2}\)
\(5^{2} \cdot x^{2}\)
\(25x^{2}\)
Answer: \(25x^{2}\)
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