Chapter Three — Basic Algebraic Expressions
The Humongous Book of Algebra Problems
51
3.39 Simplify the expression: 2 ÷ 5 + [–10 ⋅ (–5 + 11)
–1
].
According to the order of operations, the grouped expression should be
simplified first. The operations in that bracketed expression should be
completed in the following order: Simplify the inner group, apply the
exponential power, and then multiply.
Of the operations that remain, division and subtraction, division should be
completed first.
Calculate the difference of the fractions using the least common denominator.
Evaluating Expressions
Replace variables with numbers
3.40 Evaluate the expression 14x + (–x)
2
if x = –2.
Substitute x = –2 into the expression.
14x + (–x)
2
= 14(–2) + (–[–2])
2
Before you address the exponent, simplify the expression within the
parentheses: –[–2] = –1 ⋅ [–2] = 2.
The bracketed
expression has a
nested group inside
that’s surrounded by
parentheses:
(–5 + 11)
–1
.
Raising some-
thing to the –1
power moves it across
the fraction bar. When
you raise an integer or
fraction to the –1 power,
it’s the same thing as
taking the reciprocal.
The reciprocal of
is
.
The LCD
is 15. The
larger of the
two denominators
is 5. The smaller
denominator doesn’t
divide evenly into
5 ⋅ 1 = 5 or 5 ⋅ 2 = 10,
but it does divide
evenly into
5 ⋅ 3 = 15.
In other
words, change all
the x’s to “–2’s.”
Chapter Three — Basic Algebraic Expressions
The Humongous Book of Algebra Problems
52
3.41 Calculate the value of m in the formula if A = 7 and B = –28.
Substitute A and B into the formula.
Divide the numerator and denominator by 7 to reduce the fraction to lowest
terms.
7÷7
28÷7
3.42 Evaluate the expression if x = 1 and y = –4.
Substitute the given values of x and y into the expression, simplify the
numerator and denominator separately, and reduce the fraction to lowest terms.
3.43 Evaluate the expression b
2
– 4ac if a = –2, b = –3, and c = 5.
Substitute the given values of a, b, and c into the expression.
b
2
– 4ac = (–3)
2
– 4(–2)(5)
According to the order of operations, you should begin by simplifying the
exponential expression: (–3)
2
= (–3)(–3) = 9.
(–3)
2
– 4(–2)(5) = 9 – 4(–2)(5)
A negative times
a negative is positive,
so both negative signs
vanish in the next
step.
Follow the
order of operations—
multiply before
you subtract in the
numerator and
denominator.
Chapter Three — Basic Algebraic Expressions
The Humongous Book of Algebra Problems
53
Multiplication precedes subtraction in the order of operations, and this
expression contains two instances of multiplication. Work from left to right,
beginning with –4(–2) = 8.
3.44 Evaluate the expression (a + b)(a
2
– ab + b
2
) if and .
Substitute the given values of a and b into the expression.
The right factor contains exponential expressions, which should be simplified
next: and .
According to the order of operations, multiplication should be completed
before addition and subtraction: .
Combine the fractions in each set of parentheses independently. The least
common denominator of the left group is 4, and the LCD of the right group
is 16.
The square of
any real number
is positive.
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