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The Humongous Book of Algebra Problems by W. Michael Kelley

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Chapter Thirteen — Radical Expressions and Equations
The Humongous Book of Algebra Problems
290
Complex Numbers
Numbers that contain i, which equals
13.37 Simplify the expression: .
Write –49 as the product of –1 and a perfect square.
Radical expressions with an even index that contain negative radicands
represent imaginary numbers; they are defined using the value .
13.38 Simplify the expression: .
Identify the largest factor of –128 that is a perfect square to simplify the radical
expression.
Substitute into the expression and simplify.
13.39 Simplify the expression: i
13
.
If , then . Express i
13
as a product of i
2
factors.
You cant
take the square
root of a negative,
because nothing
times itself equals a
negative number. That
means negative square
roots (and fourth roots,
sixth roots,…all even
roots) are imaginary
numbers.
How can
i
2
= –1?
Usually, it’s
impossible to
square something
and get a
negative value.
However, i is an
imaginary number,
so it’s not governed
by the same
rules real
numbers
are.
There are
six –1’s here. That’s
three pairs of (1)(1):
Chapter Thirteen — Radical Expressions and Equations
The Humongous Book of Algebra Problems
291
Note: Problems 13.40–13.42 refer to complex numbers a = 2 – 3i and b = –4 + 5i.
13.40 Calculate a + b, the sum of the complex numbers, and ab, the difference of
the complex numbers.
Adding and subtracting complex numbers is very similar to adding and
subtracting polynomialscombine like terms.
To calculate ab, multiply b by –1.
Note: Problems 13.40–13.42 refer to complex numbers a = 2 – 3i and b = –4 + 5i.
13.41 Calculate a · b.
Multiply complex numbers the same way you would multiply two binomials.
According to Problem 13.39, . Substitute that value into the expression.
If youre not
sure how to multiply
binomials, look at
Problem 11.19.
Chapter Thirteen — Radical Expressions and Equations
The Humongous Book of Algebra Problems
292
Note: Problems 13.40–13.42 refer to complex numbers a = 2 – 3i and b = –4 + 5i.
13.42 Calculate a ÷ b.
Express the quotient as a fraction.
Multiply the numerator and denominator of the fraction by the conjugate of the
denominator.
Calculate the products in the numerator and denominator.
Substitute i
2
= –1 into the expression and simplify.
Complex numbers are usually expressed in the form c + di, so rewrite the
expression as a sum of two fractions with denominator 9.
The conjugate
of –4 + 5i is –4 – 5i.
All you do is change
the middle sign to
its opposite and leave
everything else alone.
Heres the benet: a
complex number times
its conjugate always
produces a real
number with no is
in it.
Each
term in the
numerator
becomes its own
fraction. Both of
the new fractions
have the same
denominator
as the big
fraction
(41).
Chapter Thirteen — Radical Expressions and Equations
The Humongous Book of Algebra Problems
293
13.43 Calculate the product: (10 + i)(6 – i).
Apply the technique described in Problem 13.41.
13.44 Calculate the quotient: .
Multiply the numerator and denominator by the conjugate of 1 – 2i.
Substitute i
2
= –1 into the expression.

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