What is 2x squared

What is 2x squared DEFAULT

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Algebra Topics: Exponents

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What are exponents?

Exponents are numbers that have been multiplied by themselves. For instance, 3 · 3 · 3 · 3 could be written as the exponent 34: the number 3 has been multiplied by itself 4 times.

Exponents are useful because they let us write long numbers in a shortened form. For instance, this number is very large:

1,,,,,,

But you could write it this way as an exponent:

1018

It also works for small numbers with many decimal places. For instance, this number is very small but has many digits:

It also could be written as an exponent:

10

Scientists often use exponents to convey very large numbers and very small ones. You'll see them often in algebra problems too.

Understanding exponents

As you saw in the video, exponents are written like this: 43 (you'd read it as 4 to the 3rd power). All exponents have two parts: the base, which is the number being multiplied; and the power, which is the number of times you multiply the base.

4 to the third power

Because our base is 4 and our power is 3, we&#x;ll need to multiply 4 by itself three times.

43 = 4 ⋅ 4 ⋅ 4 = 64

Because 4 · 4 · 4 is 64, 43 is equal to 64, too.

Occasionally, you might see the same exponent written like this: 5^3. Don&#x;t worry, it&#x;s exactly the same number&#x;the base is the number to the left, and the power is the number to the right. Depending on the type of calculator you use&#x;and especially if you&#x;re using the calculator on your phone or computer&#x;you may need to input the exponent this way to calculate it.

Exponents to the 1st and 0th power

How would you simplify these exponents?

71 70

Don&#x;t feel bad if you&#x;re confused. Even if you feel comfortable with other exponents, it&#x;s not obvious how to calculate ones with powers of 1 and 0. Luckily, these exponents follow simple rules:

  • Exponents with a power of 1
    Any exponent with a power of 1 equals the base, so 51 is 5, 71 is 7, and x1 is x.
  • Exponents with a power of 0
    Any exponent with a power of 0 equals 1, so 50 is 1, and so is 70, x0, and any other exponent with a power of 0 you can think of.

Operations with exponents

How would you solve this problem?

22 ⋅ 23

If you think you should solve the exponents first, then multiply the resulting numbers, you&#x;re right. (If you weren&#x;t sure, check out our lesson on the order of operations).

How about this one?

x3 / x2

Or this one?

2x2 + 2x2

While you can&#x;t exactly solve these problems without more information, you can simplify them. In algebra, you will often be asked to perform calculations on exponents with variables as the base. Fortunately, it&#x;s easy to add, subtract, multiply, and divide these exponents.

Adding exponents

When you&#x;re adding two exponents, you don&#x;t add the actual powers&#x;you add the bases. For instance, to simplify this expression, you would just add the variables. You have two xs, which can be written as 2x. So, x2+x2 would be 2x2.

x2 + x2 = 2x2

How about this expression?

3y4 + 2y4

You're adding 3y to 2y. Since 3 + 2 is 5, that means that 3y4 + 2y4 = 5y4.

3y4 + 2y4 = 5y4

You might have noticed that we only looked at problems where the exponents we were adding had the same variable and power. This is because you can only add exponents if their bases and exponents are exactly the same. So you can add these below because both terms have the same variable (r) and the same power (7):

4r7 + 9r7

You can never add any of these as they&#x;re written. This expression has variables with two different powers:

4r3 + 9r8

This one has the same powers but different variables, so you can't add it either:

4r2 + 9s2

Subtracting exponents

Subtracting exponents works the same as adding them. For example, can you figure out how to simplify this expression?

5x2 - 4x2

is 1, so if you said 1x2, or simply x2, you&#x;re right. Remember, just like with adding exponents, you can only subtract exponents with the same power and base.

5x2 - 4x2 = x2

Multiplying exponents

Multiplying exponents is simple, but the way you do it might surprise you. To multiply exponents, add the powers. For instance, take this expression:

x3 ⋅ x4

The powers are 3 and 4. Because 3 + 4 is 7, we can simplify this expression to x7.

x3 ⋅ x4 = x7

What about this expression?

3x2 ⋅ 2x6

The powers are 2 and 6, so our simplified exponent will have a power of 8. In this case, we&#x;ll also need to multiply the coefficients. The coefficients are 3 and 2. We need to multiply these like we would any other numbers. 3⋅2 is 6, so our simplified answer is 6x8.

3x2 ⋅ 2x6 = 6x8

You can only simplify multiplied exponents with the same variable. For example, the expression 3x2⋅2x3⋅4y2 would be simplified to 24x5⋅y2. For more information, go to our Simplifying Expressions lesson.

Dividing exponents

Dividing exponents is similar to multiplying them. Instead of adding the powers, you subtract them. Take this expression:

x8 / x2

Because 8 - 2 is 6, we know that x8/x2 is x6.

x8 / x2 = x6

What about this one?

10x4 / 2x2

If you think the answer is 5x2, you&#x;re right! 10 / 2 gives us a coefficient of 5, and subtracting the powers (4 - 2) means the power is 2.

Raising a power to a power

Sometimes you might see an equation like this:

(x5)3

An exponent on another exponent might seem confusing at first, but you already have all the skills you need to simplify this expression. Remember, an exponent means that you're multiplying the base by itself that many times. For example, 23 is 2⋅2⋅2. That means, we can rewrite (x5)3 as:

x5⋅x5⋅x5

To multiply exponents with the same base, simply add the exponents. Therefore, x5⋅x5⋅x5 = x5+5+5 = x15.

There's actually an even shorter way to simplify expressions like this. Take another look at this equation:

(x5)3 = x15

Did you notice that 5⋅3 also equals 15? Remember, multiplication is the same as adding something more than once. That means we can think of 5+5+5, which is what we did earlier, as 5 times 3. Therefore, when you raise a power to a power you can multiply the exponents.

Let's look at one more example:

(x6)4

Since 6⋅4 = 24, (x6)4 = x24

x24

Let's look at one more example:

(3x8)4

First, we can rewrite this as:

3x8⋅3x8⋅3x8⋅3x8

Remember in multiplication, order does not matter. Therefore, we can rewrite this again as:

3⋅3⋅3⋅3⋅x8⋅x8⋅x8⋅x8

Since 3⋅3⋅3⋅3 = 81 and x8⋅x8⋅x8⋅x8 = x32, our answer is:

81x32

Notice this would have also been the same as 34⋅x32.

Still confused about multiplying, dividing, or raising exponents to a power? Check out the video below to learn a trick for remembering the rules:

/en/algebra-topics/negative-numbers/content/

Sours: https://edu.gcfglobal.org/en/algebra-topics/exponents/1/
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Nonlinear equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

2x2 - 1 = 0

Step  2  :

Trying to factor as a Difference of Squares :

 2.1      Factoring:  2x2-1 

Theory : A difference of two perfect squares,  A2 - B2  can be factored into  (A+B) • (A-B)

Proof :  (A+B) • (A-B) =
         A2 - AB + BA - B2 =
         A2- AB + AB - B2 =
         A2 - B2

Note :  AB = BA is the commutative property of multiplication.

Note :  - AB + AB equals zero and is therefore eliminated from the expression.

Check : 2  is not a square !!

Ruling : Binomial can not be factored as the
difference of two perfect squares

Equation at the end of step  2  :

2x2 - 1 = 0

Step  3  :

Solving a Single Variable Equation :

 3.1      Solve  :    2x2-1 = 0 

 Add  1  to both sides of the equation : 
                      2x2 = 1
Divide both sides of the equation by 2:
                     x2 = 1/2 = 0.500
 
 When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:  
                      x  =  ± √ 1/2  

 The equation has two real solutions  
 These solutions are  x = ±√ 0.500 = ± 0.70711 
 

Two solutions were found :

                   x = ±√ 0.500 = ± 0.70711
Sours: https://www.tiger-algebra.com/drill/2x~2-1=0/
How To Factor Polynomials The Easy Way!

Three Rules of Exponents

Lesson 13, Section 2

Back to Section 1

Rule 1. Same base

Rule 2. Power of a product

Rule 3. Power of a power

Rule 1.  Same base

"To multiply powers of the same base, add the exponents."

For example,  a2a3 = a5.

Why do we add the exponents?  Because of what the symbols mean.   Section 1.

Example 1.   Multiply  3x2· 4x5· 2x

Solution.   The problem means (Lesson 5):  Multiply the numbers, then combine the powers of x :

3x2· 4x5· 2x = 24x8

Two factors of x -- x2 -- times five factors of x -- x5 -- times one factor of x, produce a total of 2 + 5 + 1 = 8 factors of x :  x8.

Problem 1.   Multiply.  Apply the rule Same Base.

To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").
Do the problem yourself first!

  a)  5x2· 6x4  = 30x6 b)  7x3· 8x6 = 56x9
 
  c)  x· 5x4 = 5x5 d)  2x· 3x· 4x = 24x3
 
  e)  x3· 3x2· 5x = 15x6 f)  x5· 6x8y2 = 6x13y2
 
  g)   4x· y· 5x2· y3 = 20x3y4 h)  2xy· 9x3y5 = 18x4y6
 
  i)  a2b3a3b4 = a5b7 j)  a2bc3b2ac = a3b3c4
 
  k)  xmynxpyq = xm + pyn+ q l)  apbqab = ap + 1bq + 1

Problem 2.   Distinguish the following:

x· x   and   x + x.

x· x = x².   x + x = 2x.

Example 2.   Compare the following:

a)  x· x5             b)  2 · 25

Solution.

a)   x· x5 = x6

b)   2 · 25 = 26

Part b) has the same form as part a).  It is part a) with x = 2.

One factor of 2 multiplies five factors of 2  producing six factors of 2.

2 · 2 = 4 is not correct here.

Problem 3.   Apply the rule Same Base.

   a)  xx7 = x8 b)  3 · 37 = 38 c)  2 · 24· 25 = 210
 
   d)  10 · 105 = 106 e)  3x· 36x6 = 37x7

Problem 4.   Apply the rule Same Base.

  a)  xnx2 = xn + 2 b)  xnx = xn + 1
 
  c)  xnxn = x2n d)  xnx1 − n = x
 
  e)  x· 2xn − 1 = 2xn f)  xnxm = xn + m
 
  g)  x2nx2 − n = xn + 2 

Rule 2:  Power of a product of factors

"Raise each factor to that same power."

For example,  (ab)3 = a3b3.

Why may we do that?  Again, according to what the symbols mean:

(ab)3 = ab· ab· ab = aaabbb = a3b3.

The order of the factors does not matter:

ab· ab· ab = aaabbb.

Problem 5.   Apply the rules of exponents.

   a)  (xy)4 = x4y4 b)  (pqr)5 = p5q5r5 c)  (2abc)3 = 23a3b3c3
  d)   x3y2z4(xyz)5=x3y2z4· x5y5z5   Rule 2.
 
 =x8y7z9   Same Base.

Rule 3:   Power of a power

"To take a power of a power, multiply the exponents."

For example,  (a2)3 = a2 · 3 = a6.

Why do we do that?  Again, because of what the symbols mean:

(a2)3 = a2a2a2 = a3 · 2 = a6

Problem 6.   Apply the rules of exponents.

   a)  (x2)5 = x10 b)  (a4)8 = a32 c)  (107)9 = 1063

Example 3.   Apply the rules of exponents:   (2x3y4)5

Solution.   Within the parentheses there are three factors:  2,  x3, and y4. According to Rule 2 we must take the fifth power of each one.  But to take a power of a power, we multiply the exponents.  Therefore,

(2x3y4)5 = 25x15y20

Problem 7.   Apply the rules of exponents.

   a)  (10a3)4 = 10,a12 b)  (3x6)2 = 9x12
 
  c)  (2a2b3)5 = 32a10b15 d)  (xy3z5)2 = x2y6z10
 
  e)   (5x2y4)3 = x6y12 f)   (2a4bc8)6 = 64a24b6c48

Problem 8.   Apply the rules of exponents.

 a)  2x5y4(2x3y6)5  = 2x5y4· 25x15y30 = 26x20y34

 b)  abc9(a2b3c4)8  = abc9· a16b24c32 = a17b25c41

Problem 9.   Use the rules of exponents to calculate the following.

 a)   (2 · 10)4 = 24· 104 = 16 · 10, = ,

 b)   (4 · 102)3 = 43· 106 = 64,,

 c)   (9 · 104)2 = 81 · 108 = 8,,,

The powers of 10 have as many 0's as the exponent of

Example 4.   Squarex4.

Solution.   (x4)2 = x8.

To square a power, double the exponent.

Problem    Square the following.

  a)  x5 = x10 b)  8a3b6 = 64a6b12
 
  c)  −6x7 = 36x14 d)  xn = x2n

Part c) illstrates:  The square of a number is never negative.

(−6)(−6) = +   The Rule of Signs.

Problem    Apply a rule of exponents -- if possible.

   a)  x2x5 = x7,   Rule 1. b)  (x2)5 = x10,   Rule 3.
   c)   x2 + x5
 Not possible. The rules of exponents apply only to multiplication.

In summary:  Add the exponents when the same base appears twice:  x2x4 = x6.  Multiply the exponents when the base appears once -- and in parentheses: (x2)5 = x10.

Problem    Apply the rules of exponents.

   a)   (xn)n = xn· n = xn2 b)   (xn)2 = x2n

Problem    Apply a rule of exponents or add like terms -- if possible.

a)   2x2 + 3x4   Not possible. These are not like terms.

b)   2x2· 3x4 = 6x6.   Rule 1.

c)   2x3 + 3x3  = 5x3.   Like terms.  The exponent does not change.

d)   x2 + y2   Not possible.  These are not like terms.

e)   x2 + x2  = 2x2.  Like terms.

f)   x2· x2  = x4.    Rule 1

g)   x2· y3  Not possible.  Different bases.

h)   2 · 26  = 27.  Rule 1

i)   35 + 35 + 35 = 3 · 35 (On adding those like terms) = 36.

We will continue the rules of exponents in Lesson

end

Next Lesson:  Multiplying out. The distributive rule.

Back to Section 1

Table of Contents &#; Home


Copyright © Lawrence Spector

Questions or comments?

E-mail:  [email protected]


Sours: https://themathpage.com/Alg/exponentshtm

Squared 2x what is

Special Product: Definition & Formula

Special Product: Definition & Formula

Special products are the result of binomials being multiplied, or simplified further, and can be solved with ease using the FOIL method: first, outer, inner, last. Learn how to use these steps, and master binomial formulas by practicing below.

Perfect Square Binomial: Definition & Explanation

Perfect Square Binomial: Definition & Explanation

A perfect square binomial is a trinomial expression in algebra that produces a square of a binomial when factored. Learn the definition and explanation of a perfect square binomial, useful formulas, and how to factor perfect square binomials.

Zero Exponent: Rule, Definition & Examples

Zero Exponent: Rule, Definition & Examples

The zero exponent rule states that when a nonzero number is raised to the power of zero, it equals 1. Explore the zero exponent rule, learn its definition, see examples, and solve an equation with zeros as exponents.

Inductive & Deductive Reasoning in Geometry: Definition & Uses

Inductive & Deductive Reasoning in Geometry: Definition & Uses

In geometry, inductive reasoning is based on observations, while deductive reasoning is based on facts, and both are used by mathematicians to discover new proofs. Learn about the definition and uses of inductive and deductive reasoning in geometry, and discover that one type of reasoning is based on observations while the other is based on facts.

Transformations in Math: Definition & Graph

Transformations in Math: Definition & Graph

The movement of objects in the coordinate plane is referred to as transformation. Learn about the different types of transformations, their graphical representations, their application in a coordinate plane, and problem-solving involving transformations.

What are Center, Shape, and Spread?

What are Center, Shape, and Spread?

Center, shape, and spread are terms used to describe the visual representation of data distribution. Explore the definitions and examples of center, shape, and spread in this lesson.

Sours: https://study.com/academy/answer/what-is-2x-squared.html
y = 2x^2

Quadratic equations

Step by step solution :

Step  1  :

Equation at the end of step  1  :

(2x2 - x) - 11 = 0

Step  2  :

Trying to factor by splitting the middle term

      Factoring  2x2-x 

The first term is,  2x2  its coefficient is  2 .
The middle term is,  -x  its coefficient is  -1 .
The last term, "the constant", is   

Step-1 : Multiply the coefficient of the first term by the constant   2 •  =  

Step-2 : Find two factors of    whose sum equals the coefficient of the middle term, which is   -1 .

        +   1   =   
        +   2   =   -9
     -2   +   11   =   9
     -1   +   22   =   21


Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored

Equation at the end of step  2  :

2x2 - x - 11 = 0

Step  3  :

Parabola, Finding the Vertex :

       Find the Vertex of   y = 2x2-x

Parabolas have a highest or a lowest point called the Vertex .   Our parabola opens up and accordingly has a lowest point (AKA absolute minimum) .   We know this even before plotting  "y"  because the coefficient of the first term, 2 , is positive (greater than zero). 

 Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two  x -intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions. 

 Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want to be able to find the coordinates of the vertex. 

 For any parabola,Ax2+Bx+C,the  x -coordinate of the vertex is given by  -B/(2A) . In our case the  x  coordinate is     

 Plugging into the parabola formula     for  x  we can calculate the  y -coordinate : 
  y = * * - * -
or   y =

Parabola, Graphing Vertex and X-Intercepts :

Root plot for :  y = 2x2-x
Axis of Symmetry (dashed)  {x}={ } 
Vertex at  {x,y} = { ,} 
 x -Intercepts (Roots) :
Root 1 at  {x,y} = {, } 
Root 2 at  {x,y} = { , } 

Solve Quadratic Equation by Completing The Square

      Solving   2x2-x = 0 by Completing The Square .

 Divide both sides of the equation by  2  to have 1 as the coefficient of the first term :
   x2-(1/2)x-(11/2) = 0

Add  11/2  to both side of the equation :
   x2-(1/2)x = 11/2

Now the clever bit: Take the coefficient of  x , which is  1/2 , divide by two, giving  1/4 , and finally square it giving  1/16 

Add  1/16  to both sides of the equation :
  On the right hand side we have :
   11/2  +  1/16   The common denominator of the two fractions is  16   Adding  (88/16)+(1/16)  gives  89/16 
  So adding to both sides we finally get :
   x2-(1/2)x+(1/16) = 89/16

Adding  1/16  has completed the left hand side into a perfect square :
   x2-(1/2)x+(1/16)  =
   (x-(1/4)) • (x-(1/4))  =
  (x-(1/4))2
Things which are equal to the same thing are also equal to one another. Since
   x2-(1/2)x+(1/16) = 89/16 and
   x2-(1/2)x+(1/16) = (x-(1/4))2
then, according to the law of transitivity,
   (x-(1/4))2 = 89/16

We'll refer to this Equation as  Eq. #  

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of
   (x-(1/4))2  is
   (x-(1/4))2/2 =
  (x-(1/4))1 =
   x-(1/4)

Now, applying the Square Root Principle to  Eq. #  we get:
   x-(1/4) = √ 89/16

Add  1/4  to both sides to obtain:
   x = 1/4 + √ 89/16

Since a square root has two values, one positive and the other negative
   x2 - (1/2)x - (11/2) = 0
   has two solutions:
  x = 1/4 + √ 89/16
   or
  x = 1/4 - √ 89/16

Note that  √ 89/16 can be written as
  √ 89  / √ 16   which is √ 89  / 4

Solve Quadratic Equation using the Quadratic Formula

      Solving    2x2-x = 0 by the Quadratic Formula .

 According to the Quadratic Formula,  x  , the solution for   Ax2+Bx+C  = 0  , where  A, B  and  C  are numbers, often called coefficients, is given by :
                                     
            - B  ±  √ B2-4AC
  x =   ————————
                      2A

  In our case,  A   =     2
                      B   =    -1
                      C   =  

Accordingly,  B2  -  4AC   =
                     1 - () =
                     89

Applying the quadratic formula :

               1 ± √ 89
   x  =    —————
                    4

  √ 89   , rounded to 4 decimal digits, is  
 So now we are looking at:
           x  =  ( 1 ±  ) / 4

Two real solutions:

 x =(1+√89)/4=

or:

 x =(1-√89)/4=

Two solutions were found :

  1.  x =(1-√89)/4=
  2.  x =(1+√89)/4=
Sours: https://www.tiger-algebra.com/drill/2x~2-x=0/

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