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## Simplifying Polynomials

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Sign in or Open in Steam. Includes 92 Steam Achievements. Publisher: Milestone S. Share Embed. Read Critic Reviews. Add to Cart. Bundle info. Add to Account.In arithmetic and algebrathe cube of a number n is its third powerthat is, the result of multiplying three instances of n together. The cube is also the number multiplied by its square :.

It is an odd functionas. The volume of a geometric cube is the cube of its side length, giving rise to the name. The inverse operation that consists of finding a number whose cube is n is called extracting the cube root of n. It determines the side of the cube of a given volume. It is also n raised to the one-third power. The graph of the cube function is known as the cubic parabola.

Because cube is an odd function, this curve has a center of symmetry at the origin, but no axis of symmetry.

A cube numberor a perfect cubeor sometimes just a cubeis a number which is the cube of an integer. Geometrically speaking, a positive integer m is a perfect cube if and only if one can arrange m solid unit cubes into a larger, solid cube. There is no minimum perfect cube, since the cube of a negative integer is negative. Unlike perfect squaresperfect cubes do not have a small number of possibilities for the last two digits. Except for cubes divisible by 5, where only 2575 and 00 can be the last two digits, any pair of digits with the last digit odd can be a perfect cube.

With even cubes, there is considerable restriction, for only 00o 2e 4o 6 and e 8 can be the last two digits of a perfect cube where o stands for any odd digit and e for any even digit. This happens if and only if the number is a perfect sixth power in this case 2 6.

It is, however, easy to show that most numbers are not perfect cubes because all perfect cubes must have digital root 18 or 9.

Moreover, the digital root of any number's cube can be determined by the remainder the number gives when divided by Every positive integer can be written as the sum of nine or fewer positive cubes. This upper limit of nine cubes cannot be reduced because, for example, 23 cannot be written as the sum of fewer than nine positive cubes:.

The smallest such integer for which such a sum is not known is Only primitive solutions are selected since the non-primitive ones can be trivially deduced from solutions for a smaller value of n.Enter an equation along with the variable you wish to solve it for and click the Solve button.

In this chapter, we will develop certain techniques that help solve problems stated in words.

### Express 81 = 3^x as a logarithmic equation?

These techniques involve rewriting problems in the form of symbols. For example, the stated problem. We call such shorthand versions of stated problems equations, or symbolic sentences.

The terms to the left of an equals sign make up the left-hand member of the equation; those to the right make up the right-hand member. The value of the variable for which the equation is true 4 in this example is called the solution of the equation. We can determine whether or not a given number is a solution of a given equation by substituting the number in place of the variable and determining the truth or falsity of the result. Solution We substitute the value 3 for x in the equation and see if the left-hand member equals the right-hand member.

The first-degree equations that we consider in this chapter have at most one solution. The solutions to many such equations can be determined by inspection.

In Section 3. However, the solutions of most equations are not immediately evident by inspection. Hence, we need some mathematical "tools" for solving equations. In solving any equation, we transform a given equation whose solution may not be obvious to an equivalent equation whose solution is easily noted. The following property, sometimes called the addition-subtraction propertyis one way that we can generate equivalent equations.

If the same quantity is added to or subtracted from both members of an equation, the resulting equation is equivalent to the original equation. The next example shows how we can generate equivalent equations by first simplifying one or both members of an equation. We want to obtain an equivalent equation in which all terms containing x are in one member and all terms not containing x are in the other. If we first add -1 to or subtract 1 from each member, we get.

In the above example, we can check the solution by substituting - 3 for x in the original equation. The symmetric property of equality is also helpful in the solution of equations. This property states.

### Cube (algebra)

This enables us to interchange the members of an equation whenever we please without having to be concerned with any changes of sign. There may be several different ways to apply the addition property above. Sometimes one method is better than another, and in some cases, the symmetric property of equality is also helpful. In this case, we get. The solution to this equation is 4. Also, note that if we divide each member of the equation by 3, we obtain the equations.

In general, we have the following property, which is sometimes called the division property. If both members of an equation are divided by the same nonzero quantity, the resulting equation is equivalent to the original equation.

In solving equations, we use the above property to produce equivalent equations in which the variable has a coefficient of 1. In the next example, we use the addition-subtraction property and the division property to solve an equation. The solution to this equation is Also, note that if we multiply each member of the equation by 4, we obtain the equations.Python is an easy to learn, powerful programming language. It has efficient high-level data structures and a simple but effective approach to object-oriented programming.

The same site also contains distributions of and pointers to many free third party Python modules, programs and tools, and additional documentation. Python is also suitable as an extension language for customizable applications.

This tutorial introduces the reader informally to the basic concepts and features of the Python language and system. It helps to have a Python interpreter handy for hands-on experience, but all examples are self-contained, so the tutorial can be read off-line as well.

For a description of standard objects and modules, see The Python Standard Library. The Python Language Reference gives a more formal definition of the language. There are also several books covering Python in depth.

This tutorial does not attempt to be comprehensive and cover every single feature, or even every commonly used feature. After reading it, you will be able to read and write Python modules and programs, and you will be ready to learn more about the various Python library modules described in The Python Standard Library.

The Glossary is also worth going through. Whetting Your Appetite. Whetting Your Appetite 2. Using the Python Interpreter 2. Invoking the Interpreter 2. Argument Passing 2. Interactive Mode 2. The Interpreter and Its Environment 2. Source Code Encoding 3. An Informal Introduction to Python 3. Using Python as a Calculator 3. Numbers 3.Not available for purchase online. Please visit a Menards store for information and to purchase.

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You'll still find the solution using algebra, but they'll be wanting a decimal approximation for non-"nice" values, which will require "technology". An example would be:. The base of the natural logarithm is the number e which has a value of about 2. This equation has a strictly numerical term being the 3 on the right-hand side. So, to solve this, I'll use The Relationship to convert the log equation to its corresponding exponential form, keeping in mind that the base of this log is " e ":.

Natural Logarithm Equations. The last value above, the cube of eis a valid solution, and often this will be all that I'm supposed to give for the answer. However, in this case maybe leading up to graphing or word problems they want me to provide a decimal approximation.

So I plug the expression into my calculator, and round the result on my screen. My answer is:. Note that this decimal form is not "better" than e 3 ; actually, e 3 is the exact, and therefore the more correct, answer.

But whereas something like 2 3 can be simplified to a straight-forward 8the irrational value of e 3 can only be approximated in the calculator. Make sure you know how to operate your calculator for finding this type of solution before the next test. This equation has a strictly-numerical term. So I'll be using The Relationship to convert the log equation to the corresponding exponential form. Then I'll solve the resulting equation.

This requires a calculator for finding the approximate decimal value. After punching a few buttons and then rounding, I find that my answer is:. If the equation has a strictly numerical term, you first use log rules to combine all log terms into one, with anything numerical on the other side of the "equals" sign.

Then you use The Relationship to convert the log equation into its corresponding exponential equation, and then you may or may not meed to use your calculator to find an approximation of the exact form of the answer. If the equation has only log terms, then you use log rules to combine the log terms to get the equation into the form "log of something equals log of something else ", and then you set something equal to something elseand solve. By the way, when finding approximations with your calculator, don't round as you go along.

Instead, do all the solving and simplification algebraically; then, at the end, do the decimal approximation as one possibly long set of commands in the calculator.

Round-off error can get really big really fast with logs, and you don't want to lose points because you rounded too early and thus too much. This equation has a strictly numerical term, so I'll be using The Relationship to convert the log equation to its corresponding exponential form, followed by some algebra:. If you try to check my solution above by plugging " 7. This is due to round-off error.

That's not to say that you can't check your answers for log equation — you most certainly can, and probably should — but you'll need to keep this round-off-error difficulty in mind when checking your solutions. In other words, when you plug your decimal approximation into the original equation, you're just making sure that the result is close enough to be reasonable.

At this point, I'll need to use the change-of-base formula to convert this to something my calculator knows how to deal with. I'll use the natural log:. No, the two values are not equal, but they're pretty darned close. Allowing for round-off error, these values confirm to me that I've gotten the right answer. If, on the other hand, my solution had returned a value of, say, Expect to need to use a calculator for log-based word problems. Let's do a couple more examples. I picked these because they can help dispell some erroneous assumptions that people often make — because too many exercises work out the same way, so people assume that they'll always work out that way.In section 3 of chapter 1 there are several very important definitions, which we have used many times.

Since these definitions take on new importance in this chapter, we will repeat them. When an algebraic expression is composed of parts to be multiplied, these parts are called the factors of the expression. It is very important to be able to distinguish between terms and factors. Rules that apply to terms will not, in general, apply to factors.

When naming terms or factors, it is necessary to regard the entire expression. From now on through all algebra you will be using the words term and factor. Make sure you understand the definitions. An exponent is a numeral used to indicate how many times a factor is to be used in a product. An exponent is usually written as a smaller in size numeral slightly above and to the right of the factor affected by the exponent.

An exponent is sometimes referred to as a "power. Note the difference between 2x 3 and 2x 3. From using parentheses as grouping symbols we see that. Unless parentheses are used, the exponent only affects the factor directly preceding it. In an expression such as 5x 4 5 is the coefficientx is the base4 is the exponent. Many students make the error of multiplying the base by the exponent. When we write a literal number such as x, it will be understood that the coefficient is one and the exponent is one.

This can be very important in many operations. It is also understood that a written numeral such as 3 has an exponent of 1. We just do not bother to write an exponent of 1. Now that we have reviewed these definitions we wish to establish the very important laws of exponents. These laws are derived directly from the definitions.

## How do you solve #3^(x-1)=81#?

To multiply factors having the same base add the exponents. For any rule, law, or formula we must always be very careful to meet the conditions required before attempting to apply it.

Note in the above law that the base is the same in both factors. This law applies only when this condition is met. These factors do not have the same base. An exponent of 1 is not usually written. This fact is necessary to apply the laws of exponents. If an expression contains the product of different bases, we apply the law to those bases that are alike.

Upon completing this section you should be able to: Recognize a monomial. Find the product of several monomials. A monomial is an algebraic expression in which the literal numbers are related only by the operation of multiplication.

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