Convert the radical $$\frac{1}{3}$$ root of a number to a rational exponent.

The radical $$\frac{1}{3}$$ root of a number can be expressed as a rational exponent: @@ oot[3]{x} = x^{rac{1}{3}}@@.

Simplify $$x^{\frac{1}{2}} \times x^{\frac{1}{3}}$$ .

Using the Product of Powers Property, $$x^{\frac{1}{2}} \times x^{\frac{1}{3}} = x^{\frac{1}{2} + \frac{1}{3}} = x^{\frac{5}{6}}$$ .

Simplify $$\frac{x^{\frac{5}{6}}}{x^{\frac{1}{2}}}$$ .

Using the Quotient of Powers Property, $$\frac{x^{\frac{5}{6}}}{x^{\frac{1}{2}}} = x^{\frac{5}{6} - \frac{1}{2}} = x^{\frac{1}{3}}$$ .

What is the power of a power property?

The Power of a Power Property states that when raising a power to another power, you multiply the exponents: $$(a^m)^n = a^{m \times n}$$ .

Simplify $$igg(x^{\frac{1}{2}}igg)^{3}$$ .

Using the Power of a Power Property, $$igg(x^{\frac{1}{2}}igg)^{3} = x^{\frac{1}{2} \times 3} = x^{\frac{3}{2}}$$ .

Convert @@ oot[5]{6^{17}}@@ to an expression with a rational exponent.

@@ oot[5]{6^{17}} = 6^{rac{17}{5}}@@.

What is the relationship between rational exponents and radicals?

Rational exponents provide a way to express roots as powers. For example, @@ oot[n]{a} = a^{rac{1}{n}}@@.

Simplify $$\frac{x^{\frac{1}{3}}}{x^{-\frac{1}{4}}}$$ .

Using the Quotient of Powers Property, $$\frac{x^{\frac{1}{3}}}{x^{-\frac{1}{4}}} = x^{\frac{1}{3} - (-\frac{1}{4})} = x^{\frac{1}{3} + \frac{1}{4}} = x^{\frac{7}{12}}$$ .

Simplify @@ oot[3]{xy}^{12}@@.

Using the Power of a Power Property, @@ oot[3]{xy}^{12} = (xy)^{rac{12}{3}} = (xy)^{4} = x^{4}y^{4}@@.

What is the significance of the denominator in a rational exponent?

The denominator of a rational exponent indicates the root to be taken. For example, in $$x^{\frac{m}{n}}$$ , the denominator $$n$$ indicates the nth root.

2.5 Properties of Rational Exponents & Radicals Flashcards

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1.

FLASHCARD QUESTION

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Define rational exponent.

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2.

FLASHCARD QUESTION

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3.

FLASHCARD QUESTION

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This property is known as the Product of Powers Property, which states that when multiplying two expressions with the same base, you can add the exponents.

4.

FLASHCARD QUESTION

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5.

FLASHCARD QUESTION

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@@x^{rac{3}{4}} = oot[4]{x^3}@@.

6.

FLASHCARD QUESTION

Front

What is the Quotient of Powers Property?

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7.

FLASHCARD QUESTION

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