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Rule 1: Adding and Subtracting
 Terms that contain exponents may only be added or subtracted if
 they are like terms.
72 ^{x+y} + 72 ^{x+y} = 2(72 ^{x+y)}

Rule II: Multiplication
Terms that contain exponents may only be multiplied if they share the same base or the same exponent.

Rule II: Multiplication
Same Bases
• Terms that share the same base can be multiplied by adding their exponents.
7^{x} ×7^{2x} =7^{3x}

Rule II Multiplication Same Exponents

• Terms that share the same exponent can be multiplied by multiplying their bases.
7^{y5} × 3^{y5} = 21^{(y2)5}

Rule II Multiplication
Same bases Same Exponents
• Terms that share the same exponent and the same exponent can be multiplied either by adding their exponents OR by multiplying their bases!
5^{y5} × 5^{y5} =25^{y10}or 25^{(y2)5}

Rule III: Division
Terms that contain exponents may only be divided if they share the same base or the same exponent.

Rule III: Division
Same Bases
 • Terms that share the same base can be divided by subtracting their exponents.
 5^{8} ÷5^{2} =5^{6}
 the bases should not be divided. Note, however, that their co
 efficients should be,the coefficients of dissimilar bases should be divided

Rule III: Division
Same Exponents
Terms that share the same exponent can be divided by dividing their bases.
x^{7} ÷y^{7} = (x/y)^{7}

Rule III: Division
Same Bases and Same Exponents
• Terms that share the same exponent and the same exponent always equal the quotient of their coefficients times 1, since any term divided by itself equals 1!
4^{5}÷4^{5}= 4^{5 }/ 4^{5} =1

Rule IV: Negative Exponents
Flip the base!
• Any term with a negative exponent can be rewritten by flipping the base and making the exponent positive.
If the negative exponent is contained in the denominator, flip the base into the numerator.
Be sure to leave the coefficients unflipped!
2x^{–2} = 2/x^{2}

Rule V: The Powers of One and Zero –
 1. Any term raised to the first power is known as a power of one,and any term raised to the zero power is known as a power of zero. ␣␣␣␣
 2. Powers of One• Any term to the first power is equal to itself.
 3. Any term to the zero power has a value of one, save for zero itself!

Rule VI: Resolving Parentheses – Before resolving the parentheses of an exponential expression, first determine whether the given term is simple or complex.
 Resolving Simple Expressions •
 To resolve the parentheses of a simple expression, distribute the exponent outside the parentheses to each term within the parentheses.
 Resolving Complex Expressions
 • To resolve the parentheses of a complex expression, combine the terms with␣in the parentheses and then distribute the exponent.

Rule VII: Consecutive Exponents –
 Multiply! Introductory
 • Any expression that contains exponents insides and outside its parentheses can be rewritten by multiplying the exponents. • Forexample,(x^{3)4} =x^{12},since3×4=12

Rule VIII: Fractioned Exponents
 To simplify an expression with a fractioned exponent:
 (1) First look at the bottom of the fraction. The denominator of a fractioned exponent indicates what root to take of the original base.
 (2) Then look at the top of the fraction. The numerator of a fractioned exponent indicates the power to which the new base should be raised.



Different Bases, Different Exponents – To solve problems that contain exponential expressions on both sides of an equation:
 1. Break down the numbers to the same base.
 2. Resolve any resulting exponent rules.
 3. Set the exponents equal to one another

Same Exponents, Different Bases –
To solve exponential equations or inequalities that contain terms with the same exponents but different bases, drop the exponents and set the bases equal to one another.



