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Level 281

### Properties of Exponents

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2⁻¹
½
2⁻³
2⁻²
¼
2⁰
1
3⁻¹
(-1)²³
-1
(-1)³²
1
-5²
-25
(-5)²
25
5⁻²
1/25
3⁻²
1/9
(-2)⁰
1
-2⁰
-1
1⁻⁵
1
-1⁵
-1
2⁻⁴
1/2⁴
2⁻⁵
1/32
x⁵⋅x
x⁶
p³/p⁷
p⁻⁴ or 1/p⁴
(2a³)(5a⁶)
10a⁹
(−c⁴)(3c²)
-3c⁶
(−x⁷)(x⁻²)
-x⁵
(−4m)(−3m)
12m²
p⁷/p³
p⁴
c⁹⋅c⁻⁴
c⁵
3u²⋅8u⁶
24u⁸
c⁻⁹⋅c⁴
Either c⁻⁵ or 1/c⁵
a⁷/a⁷
1
a⁶⋅a⁶
a¹²
a⁷⋅a⁻⁷
1
(−3d)(5d⁵)(−d³)
15d⁹
c⁶/c³
c³/c⁶
Either c⁻³ or 1/c³
1⁻⁸
1
(-1)⁸
1
0⁴
0
5/x
Express without using a negative exponent: 5x⁻¹
1/(5x)
Express without using a negative exponent: (5x)⁻¹
4/n²
Express without using a negative exponent: 4n⁻²
1/(16n²)
Express without using a negative exponent: (4n)⁻²
(3x³)²
9x⁶
(2a³)⁵
32a¹⁵
(-10p³)⁴
10,000p¹²
(-10p⁴)³
-1000p¹²
(-x²y³)⁵
-x¹⁰y¹⁵
(-x²y)⁴
x⁸y⁴
(-3w³)²
9w⁶
(-3w²)³
-27w⁶
(½)⁻³
8
(⅜)⁻²
64/9
(¾)⁻³
64/27
(1/10)⁻⁵
10,000
(10xy⁵)/(-2x⁸y²)
Either -5x⁻⁷y³ or -5y³/x⁷
(3a²b)/(6ab)
a/2
(5mn)/(10m²n³)
Either 1/(2mn²) or ½ m⁻¹n⁻²
(-3)⁻²
1/9
(-10)⁻³
-1/1000
2x²
3x² - x²
2x³y² + 5x⁵y⁶
Can't simplify further
16x²y³
5x²y³ + 11x²y³
7a + 7a²
Can't simplify further
(2a³)³/(4a)
2a⁸
Product of Powers Property
Example: 2² * 2³ = 2²⁺³
Example: (2³)² = 2³*²
Power of a Power Property
Power of a Product Property
(ab)^x = a^x * b^x
Negative Exponent Property
a non zero base raised to a negative exponent is equal to the reciprocal of the base raised to the oppositive positive exponent. a^-m = 1/a^m
Zero Exponent Property
any number raised to the zero power = 1
Quotient of Powers Property
to divide powers WITH THE SAME NON ZERO BASE, subtract the exponents a^m/a^n = a^m-n
Power of a Quotient Property
(a/b)^m = (a^m) / (b^m) when b is not zero
Linear Function
y = mx + b
Exponential Function
y = ab^x
power
a number produced by raising a base to an exponent
Expanded Form of Exponents
Expanded form utilizes the fact that exponents represent how many times the base is being multiplied. y to the 5th power (y^5) in expanded form is y*y*y*y*y.
Power Rule of Exponents
When multiplying expressions with the same base, you can add the exponents. For example, (x^2)*(x^4) = x^6. This works because you have (x*x)(x*x*x*x) which is 6 x's all multiplied.
Negative Power Rule for Exponents
When taking a number or variable to a negative exponent, it is the opposite of multiplication, which means we divide. For example, 5^-3 is the same as 1/5^3, which is 1/125. We divide by 5 three times.
Negative Power Rule for Fractions
When we take a fraction to a negative exponent, the numerator switches from multiplication to become division on the denominator. The denominator likewise turns into multiplication on the numerator. For example, (3/4)^-3 turns into (4^3)/(3^3) = 64/27.
Zero Rule for Exponents
Anything to the 0 exponent is 1. This is because we are neither multiplying or dividing by the term. For example, (3xy)^0 = 1.
Rule for Fractional Exponents
A fractional exponent can be rewritten into radical form. The root of the radical is the same as the denominator of the fraction. For example, 8^(3/2) is the same as the square root of 8^3.
Express without using a negative exponent: 5x⁻¹
5/x the 5 stays where it is. move the factor that has the negative exponent
Express without using a negative exponent: (5x)⁻¹
1/(5x) the ( ) tells you the whole thing moves to the bottom
Express without using a negative exponent: 4n⁻²
4/n² the 4 has a positive 1 as a exponent. It stays on top.