Chapter Three — Basic Algebraic Expressions

The Humongous Book of Algebra Problems

40

3.7 John is exactly 4 years less than half of Annie’s age. Express John’s age in terms

of a, Annie’s age.

Annie’s age, although unknown, is deﬁned as a. Writing John’s age in terms

of Annie’s age simply writing John’s age using the variable a. Express “half of

Annie’s age” as a product or as a quotient . John’s 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 John’s 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 dened

by the parentheses,

everything’s just

getting multiplied

together in a big pile,

so arrange the pile any

way you want. The

best way is to stick

the x’s 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.

Start Free Trial

No credit card required