Algebra 1 Common Core Student Edition, Grade 8-9

Plan
What strategy can
you use to find the
factor to remove?
You can solve a sim pler
problem by firs t ju s t
listing the factors o f the
radicand. Then choose
the greatest perfect
square on th e list.

Rem oving Perf ect - Sq uare Fact ors

What is the simplified form of V l6 0?

VT60 = Vl6 • 10 16 is th e g r e a te s t p e rfe c t-s q u a re fa c to r o f 1 60.

= V l6 • VTo Use th e M u ltip lic a tio n Property o f Square Roots.

= 4VTo Simplify V l 6.

Go t I t? 1. What is the simplified form of V72?

Sometimes you can simplify radical expressions that contain variables. A variable with

an even exponent is a perfect square. A variable with an odd exponent is the product of

a perfect square and the variable. For example, n3 = n2 • n, so Va? = Vn2 • n. In this

lesson, assume that all variables in radicands represent nonnegative numbers.

H o w is th is p ro b le m

s im ila r to P ro b le m 1?

In b o th problem s, you

need to remove a

perfect-square factor

fro m th e radicand. In

this problem, however,

th e fa c to r you remove

contains a variable.

Pr o b l em 2 Rem oving Var iab le Fact ors

Multiple Choice What is the simplified form of X' 54n7?

C£> n3 V54n GD 9«6 V6n GD 3n3 V 6n

V54n7 = V9n6 • 6 n 9ne, or (3n3)2, is a p e rfe c t-s q u a re fa c to r o f 5 4 n 7.

= V9n® • V 6 n Use the Multiplication Property of Square Roots.

= 3n3 V 6 u Simplify V9n®.

The correct answer is C.

CDC 3nVZ7n

Go t I t? 2. What is the simplified form of —m\/80m9?

You can use the Multiplication Property of Square Roots to write V a • Vb = Vflb.

Ib j.n k ——

What property allows

you to m ultiply the

whole numbers first?

The C o m m u ta tive

Property of M ultiplication

allows you to change the

order of the factors.

Pr o b l em 3 M ult iplying Tw o Rad ical Ex p r essio ns

What is the simplified form of 2 V^7t • 3 V 1 4 t2?

2 V7 1 • 3 Vl4? = 6 V7f • 14t2

= 6V98?

Multiply the whole numbers and use the

Multiplication Property of Square Roots.

= 6V49 12 • 2 1

= qVaqi? • V2 1

= 6 • 7tV2t

= 42tV2t

Simplify under the radical symbol.

49f2, o r ( 7 f ) 2, is a p e rfe c t-s q u a re fa c t o r o f 9 8 f 3.

Use th e M u ltip lic a tio n Property o f Square Roots.

Simplify V49f2.

Simplify.

620 Chapter 10 Rad i c al Exp r e ssi o n s an d Eq u at i o n s