Ch. 6_Roots Strategy

MEMORIZE both algebraic rules and example expressions

So you can imitate to simplify more complicated expressions.

Sometimes there are two numbers inside a radical sign. In order to simply this type of root

Split up the numbers into two roots then solve.

When can you simplify roots?

You can only simplify roots by combining or separating them in MULTIPLICATION AND DIVISION. *You cannot combine or separate roots in addition and subtraction.

When you decide to unsquare an equation with even exponents,

You must consider both positive AND negative solutions. E.g. x^2 = 4 has two solutions: x = 2 and x = -2

Landmark roots √144 = 12 √169 = 13 Thus, √150 is...

between 12 and 13

Landmark roots √25 = 5 √36 = 6 Thus, √30 is...

between 5 and 6

MEMORIZE: Squares and Square Roots 1^2 = 1 1.4^2 ~ 2 1.7^2 ~ 3 2.25^2 ~ 5

√(1) = 1 √(2) = 1.4 √(3) = 1.7 √(5) = 2.25

n(√x) * n(√y)

√(10) * √(5) = √(50) OR ∛(24) * ∛(9) = ∛(216) = 6

MEMORIZE: Squares and Square Roots 10^2 = 100

√(100) = 10

MEMORIZE: Squares and Square Roots 11^2 = 121

√(121) = 11

MEMORIZE: Squares and Square Roots 12^2 = 144

√(144) = 12

MEMORIZE: Squares and Square Roots 13^2 = 169

√(169) = 13

MEMORIZE: Squares and Square Roots 14^2 = 196

√(196) = 14

MEMORIZE: Squares and Square Roots 15^2 = 225

√(225) = 15

MEMORIZE: Squares and Square Roots 5^2 = 25

√(25) = 5

You can do: √(16+9)?

√(25) = 5

MEMORIZE: Squares and Square Roots 16^2 = 256

√(256) = 16

MEMORIZE: Squares and Square Roots 6^2 = 36

√(36) = 6

MEMORIZE: Squares and Square Roots 2^2 = 4

√(4) = 2

MEMORIZE: Squares and Square Roots 4^2 = 16

√(4) = 2

MEMORIZE: Squares and Square Roots 20^2 = 400

√(400) = 20

MEMORIZE: Squares and Square Roots 7^2 = 49

√(49) = 7

MEMORIZE: Squares and Square Roots 25^2 = 625

√(625) = 25

MEMORIZE: Squares and Square Roots 8^2 = 64

√(64) = 8

MEMORIZE: Squares and Square Roots 9^2 = 81

√(81) = 9

MEMORIZE: Squares and Square Roots 3^2 = 9

√(9) = 3

MEMORIZE: Squares and Square Roots 30^2 = 900

√(900) = 30

Strategy: Roots can be multiplied together.

√3(√12) = √36 = 6

Strategy Roots can be divided

√50/√2 = √25 = 5

The most common type of square root:

√64

Strategy A big number can be divided up. √98

√98 = √49 x √2 = 7√2

MEMORIZE: Cubes and Cube Roots 1^3 = 1

∛(1) = 1

MEMORIZE: Cubes and Cube Roots 5^3 = 125

∛(125) = 5

MEMORIZE: Cubes and Cube Roots 3^3 = 27

∛(27) = 3

MEMORIZE: Cubes and Cube Roots 4^3 = 64

∛(64) = 4

MEMORIZE: Cubes and Cube Roots 2^3 = 8

∛(8) = 2

broot(x^a) = (broot x)^a = x^(a/b)

25^(3/2) = √(25^3) = 5^3 = 125 OR 49^(-1/2) = 1/√(49) = 1/7 OR 5root(x^15) = x^(15/5) = x^3

√x^2y^3 + 3x^2y^3, assuming x and y are positive.

2xy√y. I said 2√x^2y^3. I didn't factor out the x^2 and y^2. We can add like terms together if they are under the same radical. Factor out all the squares and isolate them under their own radical sign. Note: √x^2 = x and √y^2 = y

What is (1/8)^(-4/3)?

(1/8)^(-4/3) = 8^(4/3) = ∛8^4 = (∛8)^4 = 2^4 = 16 Because the exponent is negative, we must take the reciprocal of the base (1/8) and change the exponent to its positive equivalent. Then we must take the 3rd (cube) root to the 4th power.

MEMORIZE: Squares and Square Roots

*These often appear on the GMAT

How can you simplify division of roots?

-You can split a larger quotient into the divide and divisor. -You can also combine two roots that are being divided into a single root of the quotient. E.g. √(144/16) = √(144) / √(16) = 12/4 = 3 OR √(72) / √(8) = √(72/8) = √(9) = 3

√2 ≅

1(1/2)

√3 ≅

1(3/4)

A root can only have a NEGATIVE value if:

1. It is an ODD root. 2. The base of the root is negative. E.g. ∛(-27) = -3

10√12 ÷2√3 = ?

10. I said 10√3. I didn't cancel the √3's. 10√12 ÷ 2√3 = 20√3/2√3 = 10

Strategy: Matching roots can be added or subtracted like variables.

3√2 + 3√2 = 7√2

∛64 equals?

4 Because 4 x 4 x 4 = 64 *We can also say that 4 is the cube root of 64.

Landmark roots √48 + √35 =

7 + 6 = 13 (a little smaller than 13).

n(√x)/n(√y)

= n(√ x/y) E.g. √(10)/√(5) = √(10/5) = √(2) OR ∛(16) / ∛(2) = ∛(16/2) = ∛(8) = 2

There is no solution for the even root of a negative number

Because no number multiplied an even number of times, can possibly be negative.

At other times, you have two roots that you would like to simplify

By combining them under one radical sign.

Estimate 4 √(5).

Combine the root and the coefficient: 4 √(5) = √(16) x √(5) = √(80) Nearest perfect square is √(81) = 9.

*Remember that you may only separate or combine the PRODUCT and QUOTIENT of two roots. *You cannot separate or combine the sum or difference of two roots.

E.g. INCORRECT: √(16 + 19) = √(16) + √(9) = 4 + 3 = 7? *You CANNOT do this.

Estimate: √(52).

Find the closest perfect square roots that you know. √(49) = 7 √(64) = 8 Thus, it will be in between 7 and 8.

You should know how to express

Fractional exponents in terms of roots and powers

What is 216^1/3?

In order to determine that root, we should break 216 in prime factors. 216 = 3 x 3 x 3 x 2 x 2 x 2 = 3^3 * 2^3 = 6^3 Therefore, the ∛216 = 6 So 216^1/3 = ∛216 = 6.

Imperfect vs perfect squares

Not all square roots are integers. E.g. √(52) does not equal an integer answer. It is an imperfect square. *No integer multiplied by itself will yield 52.

Fractional exponents are the link between roots and exponents.

Numerator = What power to raise the base to. Denominator = Which root to take. You can raise the base to the power and take the root in either order.

When we take a EVEN root (square, 4th root, 6th root), a radical sign means...

ONLY the nonnegative root of a number.

Study which errors you make carefully.

Otherwise you may find yourself trying to perform illegal manipulations on roots. *Can't separate or combine roots involving addition or subtraction.

Rule: EVEN roots only have

Positive value. E.g √4 = 2, NOT +/- 2

A root is also called

Radical

You should also know how to express

Roots as fractional exponents The resulting expression may be much easier to simplify .

You can also simplify two roots

That are being multiplied together into a single root of the product. E.g. √(25 x 16) = √(25) x √(16) = 5 x 4 = 20 OR √(50) x √(18) = √(50 x 18) = √900 = 30

CAUTION

The GMAT will try and trick you into splitting the sum or difference of two numbers inside a radical into two individual roots. The GMAT may try to trick you into combining the sum or difference of two roots inside a radical sign.

For ODD roots (cube root, 5th root, 7th root)

The root will have the SAME sign as the base. E.g. If ∛(-27) = x, what is x? X = -3 since (-3)(-3)(-3) = -27

When you see a square root symbol on the GMAT,

Think only the POSITIVE root.

Express 4root(√x) as a fractional exponent.

Transform the individual roots into exponents. √ = Exponent 1/2 4th root = Exponent 1/4 Therefore, this expression becomes (x^(1/2))^1/4 = x^1/8 = 8root(x)

It is usually better to simplify roots than to estimate them.

Unless the question asks for an estimated numerical value.

A square root has only one value

Unlike exponents, which yield both a positive and negative solution, roots have only one solution. *Since square roots have only ONE solution, they are less tricky and dangerous than EVEN exponents.

Some imperfect squares can be simplified into multiples of smaller square roots. Example: √(52)

We can rewrite √(52) as a product of primes under the radical. = √(2 x 2 x 13) - Notice the two 2's, which equals 4 (perfect square). = 2 √(13)

Simplify √(72)

We can rewrite √(72) as a product of primes. = √(2 x 2 x 2 x 3 x 3) - Notice two 2's and two 3's, which are perfect squares. Simplify into: = 2 x 3 x √(2) = 6 √(2)

If you have to estimate a root of imperfect squares

You can estimate by figuring out the two closest perfect squares on either side of it.

∛64 is an expression that answers the questions:

What number, when multiplied by itself three times, will yield 64?

How can you simplify them?

When multiplying roots, you can split up a larger product into its separate factors. Creating two separate radicals and simplifying each one individually before multiplying can save you time and having to compute large numbers.

A square root has only one value Example If √4 = x, what is x?

X = 2, since (2)(2) = 4. *GMAT follows the standard convention of math.

A root, when compared to an exponent,

is the opposite in a sense