MODULE 4

• Discipline


  Good taste<|>Excellence
• Writers: Roselyn R. Baracuso, Alden O. Madregalejo
• Cover Illustrator: Joel J. Estudillo
• This module is about simplifying expressions with rational exponents and writing expressions with rational exponents as radicals and vice versa
• Learning Objectives


  • Simplify expressions with rational exponents
  • Write expressions with rational exponents as radicals and vice versa
  • Appreciate rational exponents and radicals as applied in real life situational problems
• Simplifying expressions with rational exponents


  1. Rewrite the base in exponential form

  2. Apply the laws of exponents

  3. Simplify the result
• Writing expressions with rational exponents as radicals


  1. Use the rule: a^(x/y) = √(a^x)^(1/y)

  2. Simplify the result
• Writing expressions with radicals as rational exponents


  1. Use the rule: √(a^x)^(1/y) = a^(x/y)

  2. Simplify the result
• Rational exponents are expressions with exponents that are rational numbers
• The expressions 2^(1/2), (x^(1/3)), (x^(1/2))(y^(1/2)), and (x^4y^4)^(1/6) are examples of rational exponents
• The laws of exponents can be applied to expressions with rational exponents
• Simplifying expressions with rational exponents involves rewriting the base in exponential form, applying the laws of exponents, and simplifying the result
• To write an expression with rational exponents as a radical, use the rule: a^(x/y) = √(a^x)^(1/y)
• To write an expression with a radical as a rational exponent, use the rule: √(a^x)^(1/y) = a^(x/y)
• Simplifying expressions with rational exponents

  • 36^(1/2) = 6

  (81^(-3/4)) = 1/27

  (46)^(1/2) = 64

  (26/53)^(-1/3) = 5/4

  b^(1/3)/b^(4/3) = 1/b
• Writing expressions with rational exponents as radicals

  • 125^(4/3) = √(125^4)^(1/3) = √(15625)^(1/3) = 25

  32^(-2/5) = √(32^(-2))^(1/5) = √(1/1024)^(1/5) = 1/2

  (x^10y^20)^(-1/5) = √((x^10y^20)^(-1))^(1/5) = √(1/(x^10y^20))^(1/5) = 1/(x^2y^4)
• Writing expressions with radicals as rational exponents


  • √2 = 2^(1/2)

  √x = x^(1/2)

  √(x^2y^4) = (x^2y^4)^(1/2)
• The diagonal of a box can be found using the formula d = (l^2 + w^2 + h^2)^(1/2), where l, w, and h represent the length, width, and height of the box
• If the box is 12 cm in length, 4 cm in width, and 6 cm in height


  The length of the diagonal is (12^2 + 4^2 + 6^2)^(1/2) = 14 cm
• 0 and 21

  2
• One possible numbered-tag within 21 and 22 is 2
• 3. Is there any mathematical concept applied in the problem? Explain.
• The mathematical concept applied in the problem is all about rational exponents.
• 4. Is it possible that the position of the constituent 2^(1/2) is within these two distinct positions of the constituents 27 and 28? Explain further.
• Yes, it is because the possible numbered-tag within 27 and 28 is 2^(1/2).
• Expressions containing Rational exponents


  • x^(1/2)
  • (2x)^(2/3)
  • x^(1/4)y^(1/2)
• Expressions in radical forms


  • √2b
  • √x^(2/3)
  • √3abc^(1/4)
• a^(x/y)

  √a^x/y
• √a^(x/y)

  a^(x/y)
• What is It
• Exponent of the radicand
• Index of the radical
• Base of the radicand
• Denominator of the rational exponent
• Numerator of the rational exponent
• Base of the rational exponent
• Rational exponents
• The base a becomes the radicand.
• The index of the radical becomes the denominator of the rational exponent.
• The numerator of the exponent x becomes the exponent of the radicand.