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EXAMPLES & SOLUTIONS
Anchor 1
Without exponents, 'x + 3' would be expressed as 'x * x * x + 3'
3
Exponent expressions form terms.
There can be one or more terms to solve a exponential problem.
EVALUATE EXPRESSIONS
Explanation of expressions with exponents
evaluate expression: 2x * 3x
2
1
Follow rule of exponent
Rule: (x * x ) = x
m
n
m + n
Calculate the coefficient
coefficient: 2 * 3 = 6 --multiply rule
Determine the base
base: x
Calculate exponents
exponents: (2 + 1) = 3 --product exponent rule
Simplified Form:
6x --combine the coefficient, base & exponent
3
Evaluating Expressions
Examples of simplifying expression with exponents
Product Exponent Rule
evaluate: (x * 3x ) + 1
Rule: (x * x ) = x
1) (1 * 3) = 3 --multiply the coefficients
2) x --base
3) (2 + 1) = 3 --product rule (add) exponents
4) 3 --coefficient, base & exponent
5) 3 + 1 --simplified expression
1
m
n
m + n
2
3
x
3
x
m
Addition Exponent Rule
evaluate: ( + 3 ) + 1
Rule: (x + x ) = x
1) (1 + 3) = 4 --add the coefficients
2) --same exponent & same base
3) 4x --coefficient, base & exponent
4) 4x + 1 --simplified expression
m
m
2
x
2
x
2
x
2
2
Power of the Product Rule
evaluate: (3x * 3y)
Rule: (x * y) = x * y
1) 3 --power of the product exponents
2) 3 * 3 * 3 = 27 --multiply the coefficient 3x's
3) x & y --base
4) 27x y --coefficient, base & exponent
--simplified expression
m
m
3
m
3
3
Power of the Power Rule
evaluate: 3(x )
Rule: (x ) = x
1) (2 * 4) = 8 --power rule (multiply) exponents
2) 3 --constant
3) x --base
4) 3x --constant, base & exponent --simplified expression
m
8
n
m * n
2
4
Negative Rule
evaluate: 2x
Rule: (x) =
1) 2 --coefficient
2) x --base
3) -2 --neg. inverse rule
x = 1
4) 2
-m
-2
x
--coefficient, base, exponent
--simplified expression
m
1
2
x
2
x
( )
2y
6x
evaluate:
Power of Division Rule
2
m
y
m
m
y
( )
=
x
x
Rule:
1) 2 --division (multiply) exponent rule
2) 6 6 * 6 36
3) x
4) 36x
5) 9x
2 2 * 2 4
=
2
2
--multiply coefficients
4y
--base
2
2
2
--simplfy terms (GCF = 4)
2
--coefficients, bases, exponents
2
y
2
y
Division Exponent Rule
evaluate:
6
6x
2x
3
2
x
Rule:
x
6
=
6-3
x
1) 6 - 3 = 3 --division (subtraction) exponent rule
2) x --base
3) 6
4) 3x --coefficients, bases, exponents
--simplified expression
3
2
=
3 --divide coefficients
Fraction Rule
evaluate: 4x
=
2
3
3
4
x
2
Rule:
m
n
x
=
x
m
n
1) 2 --2 numerator - exponent
2) 3 --3 denominator - cubed root
3) 4 --coefficient
4) x -- base
3
x
2
4
--coefficient, base, exponent
--simplified expression
5)
Zero Power Rule
evaluate: 5
Rule: a = 1
1) 0 (zero) --zero power rule
2) 5 --constant and base
3) 5 = 1 --constant (base)
0
0
0
Exponent of One
evaluate: 5
Rule: a = a
1) 1 (one) --one power rule
2) 5 --constant and base
3) 5 = 5 --constant (base)
1
1
1
Examples of Combining Rules
Explanation of evaluating expression
Multiplying & Addition Rule
evaluate: 3x (x + 3x - 3 )
1) (3x * x ) + (3x * 3x ) + (3x * -3) --distributive property
2) (3 * 1) = 3 ​--multiply the coefficients
2 + 4 = 6 --multiple exponent rule
3x --updated 1st term
3) (3 * 3) = 9 --multiply the coefficients
2 + 4 = 6 --multiple exponent rule
9x --updated 2nd term
4) (3 * -3) = -9 product rule (add) exponents
x --base
-9x --updated 3rd term
5) 3x + 9x - 9x --add like terms (addition exponent)
6) 12x - 9x --updated expression
2
4
4
2
4
6
2
2
4
2
6
6
2
6
2
Power & Division Rule
evaluate:
( )
-2
x
x
5
3
1) 1 = 1 --coefficient
2) x = x --base
3) 5 - 3 = 2 --subtract
4) (1x ) --division exponent rule
5) 1 --coefficient
6) x --base
7) 2 * -2 = -4 --multiply
8) 1
x
2
-2
--power exponent rule
2
6
2
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