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RULES OF EXPONENTS
Three Laws of Exponents
1. The exponent tells how many times to multiply a number by itself.
2. The negative exponent is the opposite of multiplying so divide the number that many times.
3. The fractional exponent means to take the nth root.
- Product (or Distributive) Rule - with same base:
When multiplying powers of like bases and different exponents, keep the base and add the exponents.
𝐱 𝐚 𝐱 𝐛 = 𝐱 (𝐚+𝐛)
𝐞𝐱. 𝟐𝟑 𝟐𝟐 = 𝟐(𝟑+𝟐) = 𝟐𝟓 = 𝟑𝟐
- Product Rule - with 2 different bases & same exponents:
When multiplying powers of different bases but the same exponent,
multiply the bases and then raise the product to that exponent.
𝐱 𝐚 𝐲 𝐚 = (𝐱𝐲)𝐚
𝐞𝐱. 𝟐𝟐 𝟒𝟐 = (𝟐 ∗ 𝟒)𝟐 = 𝟖𝟐 = 𝟔𝟒 𝐎𝐑 𝟐𝟐 𝟒𝟐 = (𝟒)(𝟏𝟔) = 𝟔𝟒
- Product Rule - with 2 bases & 2 different exponents:
When multiplying powers of different bases but different exponents,
raise each base to that power and then multiply the resulting products.
It cannot be simplified, you must plug in the numbers.
𝐱𝐚𝐲𝐛 = 𝐱𝐚𝐲𝐛
𝐞𝐱. 𝟐𝟑 𝟒𝟐 = (𝟖)(𝟏𝟔) = 𝟏𝟐𝟖
- Power to Powers Rule - with 1 base and 2 exponents:
When there is one base and two exponents,
multiply the exponents and then raise the base to that power.
(𝐱 𝒂 )𝐛 = 𝐱 (𝐚∗𝐛) = 𝐱 𝐚𝐛
𝐞𝐱. (𝟐𝟐 )𝟑 = (𝟒)𝟑 𝐎𝐑 𝟐𝟐∗𝟑 = 𝟐𝟔 = 𝟔𝟒
- Power to Powers Rule - with 2 different Bases and the one exponent:
When there are two bases but only one has an exponent,
then raise that base to that power and multiply the bases.
𝐱𝐲 𝐚 = (𝐱)(𝐲)𝐚
𝐞𝐱. (𝟐 ∗ 𝟒𝟐 ) = (𝟐)(𝟒𝟐 ) = (𝟐)(𝟏𝟔) = 𝟑𝟐
- Powers to Powers Rule - with 2 Bases and 2 different exponents:
When there are two bases and two exponents,
multiply the exponents and then raise the base to that power.
𝐜
(𝐱 𝐚 𝐲 𝐛 ) = 𝐱 𝐚𝐜 𝐲 𝐛𝐜
e𝐱. (𝟐𝟐 𝟑𝟑 )𝟐 = (𝟐𝟐∗𝟐 )(𝟑𝟑∗𝟐 ) = (𝟐𝟒 )(𝟑𝟔 )
= (𝟏𝟔)(𝟕𝟐𝟗) = 𝟏𝟏𝟔𝟔𝟒
- Negative Rule: (Only move the negative exponents.)
Negative exponents in the numerator get moved to the denominator and become positive exponents.
𝒙−𝒂 = 𝟏/(𝒙𝒂 )
𝒆𝒙. 𝟐−𝟐 = 𝟏/(𝟐𝟐 ) = 𝟏/𝟒
Negative exponents in the denominator get moved to the numerator and become positive exponents.
𝟏/(𝒙−𝒂 ) = 𝒙𝒂
𝒆𝒙. 𝟏/(𝟐−𝟐 ) = 𝟐𝟐 = 𝟒
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- Quotient Rule with same base and different exponents:
When dividing two exponents with the same base: keep the base and subtract the powers.
When subtracting the powers put the answer in the numerator or denominator depending on where the
higher exponent was located to avoid negative exponents.
𝐱 𝐚 /𝐱 𝐛 = 𝐱 (𝐚−𝐛)
𝐱𝐛
and
𝐞𝐱. 𝟐𝟐 /𝟐𝟒 = 𝟐(𝟐−𝟒) = 𝟐(−𝟐) = 𝟏/(𝟐𝟐 ) = 𝟏/𝟒
= 𝐱 (𝐛−𝐚)
𝐱𝐚
𝐞𝐱.
𝟐𝟒
𝟐𝟐
= 𝟐(𝟒−𝟐) = 𝟐𝟐 = 𝟒
- Quotient Rule with different bases and the same exponent:
It cannot be simplified, you must plug in the numbers to solve.
𝐱 𝒂 /𝐲 𝒂 = (𝐱/𝐲)𝐚
𝐞𝐱. 𝟐𝟐 /𝟑𝟐 = (𝟐/𝟑)𝟐 = 𝟒/𝟗
- Quotient Rule with different bases and different exponents:
It cannot be simplified, you must plug in the numbers to solve.
𝐱𝐚
𝐲𝐛
𝐱𝐚
= 𝒚𝒃 𝐞𝐱.
𝟐𝟒
𝟑𝟐
=
𝟏𝟔
𝐱 𝐛 /𝐲 𝐚 = 𝐱 𝐛 /𝐲 𝐚 𝐞𝐱. 𝟐𝟐 /𝟑𝟒 = 𝟒/𝟖𝟏
𝟗
– Zero Rule:
Any nonzero with a zero exponent equals 1.
𝟎
Any zero with a positive exponent equals 0.
𝟎
𝟎𝒂 = 𝟎, 𝐟𝐨𝐫 𝐚 > 𝟎 𝐞𝐱. 𝟎𝟓 = 𝟎
𝐱 = 𝟏 𝐞𝐱. 𝟓 = 𝟏
Exception:
𝟎
𝟎 = 𝟎 𝐨𝐫 𝟏, 𝐭𝐡𝐞𝐫𝐞𝐟𝐨𝐫𝐞 𝐢𝐭 𝐢𝐬 𝐢𝐧𝐝𝐞𝐭𝐞𝐫𝐦𝐢𝐧𝐚𝐭𝐞
- One Rule:
𝐱 𝟏 = 𝐱 𝐞𝐱. 𝟑𝟏 = 𝟑
- Minus One Rule:
(−𝟏)𝒏
𝟏𝒂 = 𝟏 𝐞𝐱. 𝟏𝟑 = 𝟏
1 if n is even
ex.
-1 if n is odd
ex.
(−𝟏)𝟐 = 𝟏
(−𝟏)𝟓 = −𝟏
RADICAL RULES OF EXPONENTS
- Rule 𝐚
√𝐱 𝐚 𝐢𝐟 𝐚 𝐢𝐬 𝐞𝐯𝐞𝐧, 𝐭𝐡𝐞𝐧 𝐢𝐭 𝐰𝐢𝐥𝐥 𝐞𝐪𝐮𝐚𝐥 |𝐱| 𝐎𝐑 𝐢𝐟 𝐚 𝐢𝐬 𝐨𝐝𝐝, 𝐭𝐡𝐞𝐧 𝐢𝐭 𝐰𝐢𝐥𝐥 𝐞𝐪𝐮𝐚𝐥 𝐱
- Fractional Rule - Take the a-th root.
𝐱
𝟏⁄
𝐚
𝟏⁄
𝟑
𝐚
= √𝐱
𝐞𝐱. 𝟖
𝟑
= √𝟖 = 𝟐
- Fractional Rule, complicated Take the denominator (the ‘b’ power) root and then do the numerator (the ‘a’ power).
𝟒
𝒃
𝐛
𝐱 𝐚/𝐛 = ( √𝐱 )𝐚 = √𝒙𝒂
𝐞𝐱. 𝟏𝟔𝟑/𝟒 = ( √𝟏𝟔 )𝟑 = 𝟐𝟑 = 𝟖
- Product Rule –
Product of 2 roots is the root of the products.
𝐚
[( √𝐱) ∗ ( 𝐚√𝐲)] = ( 𝐚√𝐱𝐲)
𝟐
𝟐
𝟐
𝟐
𝐞𝐱. [(√𝟏𝟔) ∗ (√𝟗)] = (√𝟏𝟔 ∗ 𝟗) = [𝟒 ∗ 𝟑] = (√𝟏𝟒𝟒) = [𝟏𝟐] = (𝟏𝟐)
- Power Rule –
Index (a) of the root of a root is the product of their indexes (a*b).
𝐚 𝐛
𝐚𝐛
√ √𝐱 =
√𝐱
𝟐
𝟑
𝐞𝐱. √ √𝟔𝟒 =
𝟐∗𝟑
=
√𝟔𝟒
𝟔
√𝟔𝟒
- Quotient Rule - It cannot be simplified, you must plug in the numbers to solve.
𝐚 𝐱
√𝐲
𝐚
=
√𝐱
𝐚
√𝐲
𝐖𝐡𝐞𝐫𝐞 𝐲 ≠ 𝟎 ex.
𝟐 𝟏𝟔
√
𝟗
𝟐
=
√𝟏𝟔
𝟐
√𝟗
=
𝟒
𝟑
= 𝟐