Rational Exponents

1
Rational Exponents

• By
• Elyse Valente, Anita Caceda, Haley Sutera
• Sabrina Giuliano

2
Rational Exponents are...

• Exponents that are fractions
• and
• fractional powers
• where a number is raised to a fraction

3

• Rational Exponents have the exact same properties
  to those of the basic exponential properties.
• Properties
• 1. am an(mn)
• 2. (am)n amn
• 3. (ab)m ambm
• 4. a-m 1/am, a doesn't 0/
• 5. am / an am-n, a does not 0
• 6. (a/b)m am /
  bm, b does not 0

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Examples Using Multiplication Property

• am an a(mn)
• 31/2 33/2 3(1/2 3/2) 32 9
• By using the multiplication property, the two
  exponents were added together because the base
  was the same and the denominators were the same
  leaving the answer to be a whole number, 9.

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Examples Using Multiplication Property with
Parenthesis

• (am)n amn
• a2 m3 n4
• (23)4 2(34)
• 3 x 4 12 - 212
• - When one is given a number to a power, then
  raised to another power, this property requires
  the exponents to be multiplied together.
  Ultimately, leaving the base to be raised to the
  product of the two exponents. The base in the
  situation, 2 after this property, is raised to
  the 12th.

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(ab)m ambm

• a4 b3 m2(4x3)2 42 x 32 16 x 9 144-
  Similar to the distributive property, the
  exponent outside the parenthesis using this
  property is raised to both bases. As opposed to
  multiplying the bases then raising it to the 2nd
  power. In this case, the 2 is distributed to
  both the 4 and the 3.

7
a-m 1/am, a doesn't 0

• Whenever there is a base raised to a negative
  exponent, it is equivalent to turning it into a
  fraction. Where the exponent turns positive in
  the denominator, with 1 as the numerator.
• a4 -m 2 4-2 1/ 42 1/16
• a does NOT equal 0

8
am / an am-n, a does not 0

• Division Subtraction
• a4 m3 n2 43 / 42 4 (3-2) 41
  4Division, with rational exponents is immediataly
  linked with subtraction. When dividing rational
  exponents, it is equivalent to subtracting the
  the exponents. This can only be done when the
  bases are the same. Because the bases are both 4,
  the exponents can be subtracted.

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(a/b)m am / bm, b does not 0

• a8, b27, m1/3
• (8/27)1/3 8(1/3) / 27(1/3) 2/3
• By using the division property you are able to
  subtract the exponents, therefore 1/3 subtracted
  by 1/3 equals 0 and you are left with 8/27 which
  equals the value of 2/3.

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Frequent Mistakes

• While using the multiplication property (without
  the parenthesis) it is common for people to want
  to multiply the exponents when you are advised to
  add them. am an a(mn)
• A common mistake made using the division property
  (without the parenthesis) is usually diving the
  exponents when you are supposed to subtract them
  because the parentheses aren't there to tell you
  otherwise. am / an a(m-n)