9.7 Summary of key concepts

Elementary algebra Page 1 / 1

This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The distinction between the principal square root of the number x and the secondary square root of the number x is made by explanation and by example. The simplification of the radical expressions that both involve and do not involve fractions is shown in many detailed examples; this is followed by an explanation of how and why radicals are eliminated from the denominator of a radical expression. Real-life applications of radical equations have been included, such as problems involving daily output, daily sales, electronic resonance frequency, and kinetic energy.This module presents a summary of the key concepts of the chapter "Roots, Radicals, and Square Root Equations".

Summary of key concepts

Square root ( [link] )

The square root of a positive number

x

is a number such that when it is squared, the number

x

results.

Every positive number has two square roots, one positive and one negative. They are opposites of each other.

Principal square root $\sqrt{x}$ ( [link] )

x

is a positive real number, then

\sqrt{x}

represents the positive square root of

x

. The positive square root of a number is called the principal square root of the number.

Secondary square root $- \sqrt{x}$ ( [link] )

- \sqrt{x}

represents the negative square root of

x

. The negative square root of a number is called the secondary square root of the number.

Radical sign, radicand; and radical ( [link] )

In the expression

\sqrt{x},

√ is called the radical sign .

x

is called the radicand .

\sqrt{x}

is called a radical .

The horizontal bar that appears attached to the radical sign, √, is a grouping symbol that specifies the radicand.

Meaningful expressions ( [link] )

A radical expression will only be meaningful if the radicand (the expression under the radical sign) is not negative:

\sqrt{- 25}

is not meaningful and

\sqrt{- 25}

is not a real number

Simplifying square root expressions ( [link] )

a

is a nonnegative number, then

\sqrt{a^{2}} = a

Perfect squares ( [link] )

Real numbers that are squares of rational numbers are called perfect squares .

Irrational numbers ( [link] )

Any indicated square root whose radicand is not a perfect square is an irrational number.

\sqrt{2}, \sqrt{5}

and

\sqrt{10}

are irrational numbers

The product property ( [link] )

\begin{matrix} \sqrt{x y} & = & \sqrt{x} \end{matrix} \sqrt{y}

The quotient property ( [link] )

\begin{matrix} \sqrt{\frac{x}{y}} = \frac{\sqrt{x}}{\sqrt{y}} & y \neq 0 \end{matrix}

Be careful ( [link] )

\begin{array}{l} \sqrt{x + y} & \neq & \sqrt{x} + \sqrt{y} & (\sqrt{16 + 9} & \neq & \sqrt{16} + \sqrt{9}) \\ \sqrt{x - y} & \neq & \sqrt{x} - \sqrt{y} & (\sqrt{25 - 16} & \neq & \sqrt{25} - \sqrt{16}) \end{array}

Simplified form ( [link] )

A square root that does not involve fractions is in simplified form if there are no perfect squares in the radicand.

A square root involving a fraction is in simplified form if there are no

perfect squares in the radicand,
fractions in the radicand, or
square root expressions in the denominator

Rationalizing the denominator ( [link] )

The process of eliminating radicals from the denominator is called rationalizing the denominator .

Multiplying square root expressions ( [link] )

The product of the square roots is the square root of the product.

\sqrt{x} \sqrt{y} = \sqrt{x y}

Simplify each square root, if necessary.
Perform the multiplication.
Simplify, if necessary.

Dividing square root expressions ( [link] )

The quotient of the square roots is the square root of the quotient.

\frac{\sqrt{x}}{\sqrt{y}} = \sqrt{\frac{x}{y}}

Addition and subtraction of square root expressions ( [link] )

\begin{array}{l} a \sqrt{x} + b \sqrt{x} = (a + b) \sqrt{x} \\ a \sqrt{x} - b \sqrt{x} = (a - b) \sqrt{x} \end{array}

Square root equation ( [link] )

A square root equation is an equation that contains a variable under a square root radical sign.

Solving square root equations ( [link] )

Isolate a radical.
Square both sides of the equation.
Simplify by combining like terms.
Repeat step 1 if radical are still present.
Obtain potential solution by solving the resulting non-square root equation.
Check potential solutions by substitution.

Questions & Answers

how did you get 1640

Noor Reply

If auger is pair are the roots of equation x2+5x-3=0

Peter Reply

Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.

MATTHEW Reply

420

Sharon

from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph

George

Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28

Salma

Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.

Aaron Reply

find product (-6m+6) ( 3m²+4m-3)

SIMRAN Reply

-42m²+60m-18

Salma

what is the solution

bill

how did you arrive at this answer?

bill

-24m+3+3mÁ^2

Susan

i really want to learn

Amira

I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please

Amira

Hw did u arrive to this answer.

Aphelele

Bajemah

-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18

Salma

complete the table of valuesfor each given equatio then graph. 1.x+2y=3

Jovelyn Reply

x=3-2y

Salma

y=x+3/2

Salma

Enock

given that (7x-5):(2+4x)=8:7find the value of x

Nandala

3x-12y=18

Kelvin

please why isn't that the 0is in ten thousand place

Grace Reply

please why is it that the 0is in the place of ten thousand

Grace

Send the example to me here and let me see

Stephen

A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg

Marry Reply

how far

Abubakar

cool u

Enock

state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°

Abegail Reply

hello

BenJay

Method

I am eliacin, I need your help in maths

Rood

how can I help

Sir

hmm can we speak here?

Amoon

however, may I ask you some questions about Algarba?

Amoon

Enock

what the last part of the problem mean?

Roger

The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.

cameron Reply

Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.

mahnoor Reply

I'm guessing, but it's somewhere around $4335.00 I think

Lewis

12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00

Munster

difference between rational and irrational numbers

Arundhati Reply

When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?

Jakoiya Reply

how to reduced echelon form

Solomon Reply

Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?

Zack Reply

d=r×t the equation would be 8/r+24/r+4=3 worked out

Sheirtina

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Source: OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3

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9.7 Summary of key concepts

Summary of key concepts

Square root ( [link] )

Principal square root x ( [link] )

Secondary square root − x ( [link] )

Radical sign, radicand; and radical ( [link] )

Meaningful expressions ( [link] )

Simplifying square root expressions ( [link] )

Perfect squares ( [link] )

Irrational numbers ( [link] )

The product property ( [link] )

The quotient property ( [link] )

Be careful ( [link] )

Simplified form ( [link] )

Rationalizing the denominator ( [link] )

Multiplying square root expressions ( [link] )

Dividing square root expressions ( [link] )

Addition and subtraction of square root expressions ( [link] )

Square root equation ( [link] )

Solving square root equations ( [link] )

Questions & Answers

Read also:

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Principal square root $\sqrt{x}$ ( [link] )

Secondary square root $- \sqrt{x}$ ( [link] )