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 can’t

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 a – b, the difference of

the complex numbers.

Adding and subtracting complex numbers is very similar to adding and

subtracting polynomials—combine like terms.

To calculate a – b, 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 you’re 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.

Here’s the benet: a

complex number times

its conjugate always

produces a real

number with no i’s

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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