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Section 7 – 1A:
Expressions with Rational Exponents
The Product Rule for Mononominal Terms
(Ax C y D ) (Bx E y F ) = A • B• x C +E • y D+ F
The Product Rule for Mononominal Terms has been used in past sections with bases that were raised
to a whole number exponent or an integer exponent.
Example 1
Example 2
whole number exponents
−2x 4 y 5xy 5
Integer exponents
3x −2 y 7 2x 6 y −5
= −2 ⋅ 5x 4 +1 y 1+ 5
= 6 • x −2 + 6 y 7− 5
= −10 x 5 y 6
= 6x 4 y 2
(
)(
)
(
)(
)
TThe Product Rule for Mononominal Terms can also be used with nonnegitive bases that are raised to
a exponent that is a real number. The rule still says to combine (add or subtract) the exponents of
a common base. The exponents in these examples are fractions instead of integers.
Example 3
Example 4
x4 3• x2 3
combine (add or subtract)
the exponents for the base x
y 3 4 • y −1 2
combine (add or subtract)
the exponents for the base x
4 2
+
x3 3
3 2
−
4
4
x
x 6/3
= x1/4
= x2
Example 5
Example 6
y 2 3 • y −1 2
combine (add or subtract)
the exponents for the base x
y 1 3 • y −5 3
combine (add or subtract)
the exponents for the base x
1 5
−
x3 3
4 3
−
x6 6
=x
1/6
=x
−
4
3
1
= 4/3
x
Math 120 Section 7 – 1A
Page 1
© 2012 Eitel
The Power Rule for Mononominal Terms
where x and y are nonnegitive real numbers and A, B , C and D are real numbers
(A 1 x B y C )
D
= A1•D x B•D y C•D
The Power Rule for Mononominal Terms can be used with nonnegitive bases that are raised to a
exponent that is a real number. The rule still says to multiply the exponent of each base inside
the parentheses by the exponent outside the parentheses. The exponents in these examples
are fractions instead of integers.
Example 7
Example 8
( y −5)
(x )
9 23
13
multiply exponents
multiply exponents
−5•
1
3
2
9•
x 3
x
= x6
= x −5/3 =
Example 9
Example 10
( x 9 y −1)
(x y )
3 5 32
9•
3
5•
•y 2
x
=
=
9
x2
15
•y 2
=
Math 120 Section 7 – 1A
x 5/3
23
multiply exponents
multiply exponents
3
3•
x 2
1
Page 2
2
3
−1•
•y
2
3
−2
6 •3
x y
x6
y 2/3
© 2012 Eitel
The Quotient Rule for Mononominal terms
where x is a nonnegitive real number and T, B are real numbers
The QuotientRule for Mononominal Terms has been used in past sections with nonnegitive bases that
were raised to an exponent that was an integer. The QuotientRule for Mononominal Terms is used
when there is a common base with one base in the numerator and the other in the
denominator.
If the top exponent > bottom exponent
If the bottom exponent > top exponent
x T x T −B
=
1
xB
xT
1
= B− T
B
x
x
The Quotient Rule for Mononominal Terms can be used with nonnegitive bases that are raised to a
exponent that is a real number. The rule still says to subtract the exponent of the common
base. The exponents in these examples are fractions instead of integers.
instead of integers.
Example 13
y9 4
y1 4
subtract exponents
9 1
−
4
y 4
=
8
4
y
=y
2
Example 14
x2 3
x8 3
subtract exponents
1
8 2
−
x3 3
=
x 5 2 y −2 3
x −1 2 y 7 3
move negitive exponents
x 5 2 x1 2
y 7 3 y2 3
add exponents
5 1
+
x2 2
1
6
x3
7 2
+
y3 3
1
= 2
x
Math 120 Section 7 – 1A
Example 15
=
Page 3
6
x3
9
y3
=
x2
y3
© 2012 Eitel