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No credit card required Chapter Three — Basic Algebraic Expressions
The Humongous Book of Algebra Problems
40
3.7 John is exactly 4 years less than half of Annies age. Express Johns age in terms
of a, Annies age.
Annies age, although unknown, is deﬁned as a. Writing Johns age in terms
of Annies age simply writing Johns age using the variable a. Express “half of
Annies age” as a product or as a quotient . Johns age is 4
years less than that: or .
Exponential Expressions
Rules for simplifying expressions that contain powers
3.8 Simplify the expression and verify the result: (2
3
)(2
4
).
To calculate the product of two exponential expressions with the same base,
raise the common base to the sum of the powers: (x
a
)(x
b
) = x
a + b
.
(2
3
)(2
4
) = 2
3 + 4
= 2
7
To verify the result, expand the expressions 2
3
and 2
4
. An exponential
expression indicates that the given base is multiplied by itself repeatedly,
according to the value of the exponent. To calculate 2
3
, you must multiply three
2s together, and to calculate 2
4
, multiply four 2s.
Substitute 2
3
= 8 and 2
4
= 16 into the original expression and verify that the
product equals 2
7
.
3.9 Simplify the expression: y
3
· y
6
· y
13
· y.
The rightmost factor in this expression has no explicit exponent value, so
its implicit exponent is 1. Rewrite the expression so that each factor has a
corresponding exponent.
y
3
· y
6
· y
13
· y = y
3
· y
6
· y
13
· y
1
All the factors have common base y, so add the powers to calculate the product
(as explained in Problem 3.8).
You can also
simplify
using common
denominators and
write Johns age as a
single fraction:
When you raise
a number to a
power, that number
is called the base.
In this problem, 2
3
and 2
4
have the
same base: 2. To
multiply things with
the same base, add
the powers (3 + 4 =
7) and raise the
base to that
power: 2
7
.
If no power
is written, the
power is 1. Chapter Three — Basic Algebraic Expressions
The Humongous Book of Algebra Problems
41
3.10 Simplify the expression: (x
3
y
5
)(xy
9
).
Multiplication is commutative, so the order in which the factors are written
has no impact upon the product. Rewrite the expression by grouping the
common bases and express xy
9
as x
1
y
9
so that each factor has an explicitly stated
exponent.
To calculate the product of exponential expressions with a common base, raise
the common base to the sum of the exponents. In this case, x
3
and x
1
share a
common base, as do y
5
and y
9
.
3.11 Simplify the expression: .
To determine the quotient of two exponential expressions with a common base,
calculate the difference of the exponents and raise the common base to that
power: .
3.12 Simplify the expression: .
As explained in Problem 3.11, the quotient of two exponential expressions with
a common base is the common base raised to the difference of the exponents.
3.13 Simplify the expression: . Assume that w 0.
Rewrite the product as a single fraction by multiplying the numerators together
and dividing by the product of the denominators. State the exponents of x and z
in the denominator explicitly: x = x
1
and z = z
1
.
Apply the exponential property described in Problem 3.11: The quotient of
exponential expressions with the same base is equal to the common base raised
to the difference of the powers.
Even though
there are two
groups dened
by the parentheses,
everythings just
getting multiplied
together in a big pile,
so arrange the pile any
way you want. The
best way is to stick
the xs and y’s
together.
Let’s say you
have two things
with the same base,
such as w
6
and w
2
. To
multiply them, add the
exponents: (w
6
)(w
2
) =
w
6 + 2
= w
8
. To divide
them, subtract the
exponents:
w
6
÷ w
2
= w
6 – 2
= w
4
.
All the factors
in the top of the
fraction can be moved
around, as can the
factors in the bottom, if
the things on top stay
on top and the things
in the bottom stay
down there.

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