Chapter Thirteen — Radical Expressions and Equations

The Humongous Book of Algebra Problems

276

Simplifying Radical Expressions

Moving things out from under the radical

Note: Problems 13.1–13.2 refer to the radical expression .

13.1 Identify the radicand and index of the expression.

The index of a radical expression is the small number outside and to the left of

the radical symbol; the index of is 3. The value inside the radical symbol is

called the radicand; the radicand of is 64.

Note: Problems 13.1–13.2 refer to the radical expression .

13.2 Simplify the expression and verify your answer.

As explained in Problem 13.1, the index of is 3. To simplify the cube root,

try to identify factors of the radicand that are perfect cubes. It is helpful to

memorize the ﬁrst six perfect cubes, listed here.

Notice that 64 is a perfect cube, as 4

3

= 64. Rewrite the radical expression,

identifying the perfect cube explicitly.

Any factor of the radicand that is raised to an exponent equal to the index of

the radical can be removed from the radical.

Therefore, . To verify this solution, raise the answer (4) to an exponent

equal to the index of the original radical (3). The result should be the original

radicand (64).

4

3

= 64

Note: Problems 13.3–13.4 refer to the radical expression .

13.3 Identify the index and radicand of the expression.

The radicand of is the number inside the radical symbol: 36. No index is

written explicitly, which indicates an implied index of 2. Square roots are rarely

written with an explicit index, so the notation is preferred to .

Radicals

with index

3 are called

“cube roots,”

just like raising

something to the

third power is called

“cubing” it. Radicals

with index 2 are

called “square

roots,” just like

raising something

to the second

power is called

“squaring” it.

The 4

inside the

radical symbol

has exponent 3,

and the index of

the radical (that

tiny number oating

out front) is also 3.

When the power and

the index match, they

sort of cancel each

other out. The 3

exponent goes away,

the index goes

away, and the

radical sign

disappears,

too.

When there’s no little

number nestled in the check mark

outside the radical, that means

you’re dealing with a square root,

and you can assume the index

is 2.

Chapter Thirteen — Radical Expressions and Equations

The Humongous Book of Algebra Problems

277

Note: Problems 13.3–13.4 refer to the radical expression .

13.4 Simplify the expression and verify your answer.

According to Problem 13.3, the index of is 2. To simplify the square root,

factor the radicand using perfect squares. It is helpful to memorize the ﬁrst 15

perfect squares, listed here.

Notice that 36 is a perfect square, as 6

2

= 36. Rewrite the radical expression,

identifying the perfect square explicitly.

As stated in Problem 13.2, any factor in the radicand that is raised to a power

equal to the index of the radical can be simpliﬁed.

To verify that , raise the answer (6) to a power that equals the index of

the original radical (2). The answer must match the original radicand (36).

6

2

= 36

13.5 Simplify the radical expression: .

The index of is not written explicitly, so according to Problem 13.3, the

index is 2 and the radical expression is a square root. Consider the perfect

squares listed in Problem 13.4 and determine whether any of those values divide

evenly into the radicand.

Because dividing 50 by 25 produces no remainder (50 ÷ 25 = 2), 25 is a factor of

50. Rewrite the radicand in factored form, explicitly identifying 25 as a perfect

square.

The root of a product is equal to the product of the roots.

The exponent of 5 and the index of the radical expression are equal, so simplify

the radical expression: .

Therefore, .

The index of

the radical is 2,

and the exponent

of 6 is 2. Those 2’s

cancel out and take

the radical sign

away with them. All

that’s left is 6.

Don’t even

bother with

1

2

= 1. Sure 1 is a

perfect square, but

it divides into every

number evenly and is

not all that useful

for simplifying

radicals.

In other words,

when two things are

multiplied inside a

root (a radical symbol),

you can break them

into two separate roots

that are multiplied:

. This

does NOT necessarily

work when two things

inside radicals are

added or subtracted:

.

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