Properties of rational exponents quick check

Answer 1

Certainly! Here’s a quick overview of the properties of rational exponents:

Properties of Rational Exponents

1. Definition:

   • If \( a \) is a non-negative real number and \( m/n \) is a rational number (where \( n \neq 0 \)), then: \[ a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m \]

2. Product of Powers:

   • \( a^{m/n} \cdot a^{p/q} = a^{(mq + pn) / (nq)} \)
   • Example: \( a^{1/2} \cdot a^{1/3} = a^{(1 \cdot 3 + 1 \cdot 2) / (2 \cdot 3)} = a^{5/6} \)

3. Quotient of Powers:

   • \( \frac{a^{m/n}}{a^{p/q}} = a^{(mq - pn) / (nq)} \)
   • Example: \( \frac{a^{1/2}}{a^{1/3}} = a^{(1 \cdot 3 - 1 \cdot 2) / (2 \cdot 3)} = a^{1/6} \)

4. Power of a Power:

   • \( (a^{m/n})^p = a^{mp/n} \)
   • Example: \( (a^{1/2})^3 = a^{3/2} \)

5. Power of a Product:

   • \( (ab)^{m/n} = a^{m/n} \cdot b^{m/n} \)
   • Example: \( (xy)^{1/2} = x^{1/2} \cdot y^{1/2} \)

6. Power of a Quotient:

   • \( \left(\frac{a}{b}\right)^{m/n} = \frac{a^{m/n}}{b^{m/n}} \)
   • Example: \( \left(\frac{x}{y}\right)^{1/2} = \frac{x^{1/2}}{y^{1/2}} \)

7. Zero Exponent:

   • \( a^0 = 1 \) (where \( a \neq 0 \))

8. Negative Exponent:

   • \( a^{-m/n} = \frac{1}{a^{m/n}} \)

Example Calculations

• \( 2^{3/2} = \sqrt{2^3} = \sqrt{8} = 2\sqrt{2} \)
• \( x^{1/3} \cdot x^{2/3} = x^{(1+2)/3} = x^{3/3} = x^1 = x \)
• \( \frac{y^{2/5}}{y^{1/5}} = y^{(2-1)/5} = y^{1/5} \)

These basic properties allow you to simplify and manipulate expressions with rational exponents effectively. If you have specific questions or examples that you would like to explore further, feel free to ask!