X 5 x 5 simplify SIMPLIFY UISNG LAWS OF EXPONENTS WORKSHEET

(1)  Simplify (-7)² x (-3)³

(2)  Simplify (-2)⁵ x (-1)⁸³

(3)  Simplify (-4)⁵ ÷ (-4)⁸

(4)  Simplify (-3)⁴ x (5/3)⁴

(5)  Simplify (3⁻⁷÷ 3¹⁰) x 3⁻⁵

(6)  Simplify (7)  Simplify (8)  Simplify (9)  Simplify (10)  Simplify  Question 1 :

Simplify (-7)² x (-3)³

Solution :

Since the power of -7 is even, the answer must have positive sign. So it is enough to multiply 7 two times.

(-7)2 = 7 x 7 ==> 49

Since the power of -3 is odd, the answer will have negative sign.

(-3)3 = 3 x 3 x 3 ==> -27

(-7)2 x (-3)3 = 49 x (-27) ==> -1323

Question 2 :

Simplify (-2)⁵ x (-1)83

Solution :

(-2)⁵  =  -32

(-1)83  =  -1

=  -32(-1)

=  32

Question 3 :

Simplify (-4)⁵ ÷ (-4)⁸

Solution :

=  (-4)⁵/(-4)⁸

=  (-4) 5-8

=  (-4)-3

=  1/(-4)3

=  -1/64

Question 4 :

Simplify (-3)4 x (5/3)4

Solution :

(-3)⁴ x (5/3)⁴

=  (3)⁴ x (5⁴/3⁴)

By canceling 34 in both numerator and denominator, we get

= 5⁴ ==> 5 x 5 x 5 x 5 ==> 625

Question 5 :

Simplify (3-7÷ 310) x 3-5

Solution :

=  (3-7÷ 310) x 3-5

=  (3-7x 3-10) x 3-5

=  3-7-10-5

=  3-22

=  1/322

Question 6 :

Simplify Solution : Question 7 :

Simplify Solution : Question 8 :

Simplify Solution : Question 9 :

Simplify Solution :

= (-2)2 + (-3)2 + (-4)2

= 4 + 9 + 16 ==> 29

Question 10 :

Simplify Solution :

= (-3)4

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Remainder when 2 power 256 is divided by 17

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Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

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Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6

Sours: https://www.onlinemath4all.com/simplify-using-laws-of-exponents-worksheet.html

Simplify by Using the Product, Quotient, and Power Rules

Learning Objective(s)

·         Use the product rule to multiply exponential expressions with like bases.

·         Use the power rule to raise powers to powers.

·         Use the quotient rule to divide exponential expressions with like bases.

·         Simplify expressions using a combination of the properties.

Exponential notation was developed to write repeated multiplication more efficiently. There are times when it is easier to leave the expressions in exponential notation when multiplying or dividing. Let’s look at rules that will allow you to do this.

The Product Rule for Exponents

Recall that exponents are a way of representing repeated multiplication. For example, the notation 54 can be expanded and written as 5 • 5 • 5 • 5, or 625. And don’t forget, the exponent only applies to the number immediately to its left, unless there are parentheses.

What happens if you multiply two numbers in exponential form with the same base? Consider the expression (23)(24). Expanding each exponent, this can be rewritten as (2 • 2 • 2) (2 • 2 • 2 • 2) or  2 • 2 • 2 • 2 • 2 • 2 • 2. In exponential form, you would write the product as 27. Notice, 7 is the sum of the original two exponents, 3 and 4.

What about (x2)(x6)? This can be written as (x • x)(x • x • x • x • x • x) = x • x • x • x • x • x • x • x  or x8. And, once again, 8 is the sum of the original two exponents.

To multiply exponential terms with the same base, simply add the exponents.

 Example Problem Simplify.(a3)(a7) (a3)(a7) The base of both exponents is a, so the product rule applies. a3+7 Add the exponents with a common base. (a3)(a7) = a10

When multiplying more complicated terms, multiply the coefficients and then multiply the variables.

Example

Problem

Simplify.

5a4· 7a6

35 ·a4 ·a6

Multiply the coefficients.

35 ·a4+6

The base of both exponents is a, so the product rule applies. Add the exponents.

35 ·a10

Add the exponents with a common base.

5a4· 7a6 = 35a10

 Simplify the expression, keeping the answer in exponential notation.  (4x5)( 2x8) A) 8x5 • x8B) 6x13C) 8x13D) 8x40  Show/Hide AnswerA) 8x5 • x8Incorrect. 8x5• x8is equivalent to (4x5)(2x8), but it still is not in simplest form. Simplify x5•x8 by using the Product Rule to add exponents. The correct answer is 8x13. B) 6x13Incorrect. 6x13 is not equivalent to (4x5)(2x8). In this incorrect response, the correct exponents were added, but the coefficients were also added together. They should have beenmultiplied. The correct answer is 8x13.  C) 8x13Correct. 8x13  is equivalent to (4x5)(2x8). Multiply the coefficients (4 • 2) and apply the Product Rule to add the exponents of the variables (in this case x) that are the same. D) 8x40Incorrect. 8x40 is not equivalent to (4x5)(2x8). Do not multiply the coefficients and the exponents. Remember, using the Product Rule add the exponents when the bases are the same. The correct answer is 8x13.

The Power Rule for Exponents

Let’s simplify (52)4. In this case, the base is52 and the exponent is 4, so you multiply 52 four times:(52)4  = 52 52 •52 •52 = 58 (using the Product Rule – add the exponents).

(52)isa power of a power. It is the fourth power of 5 to the second power. And we saw above that the answer is 58. Notice that the new exponent is the same as the product of the original exponents: 2 •4 = 8.

So, (52)= 52 4 =  58 (which equals 390,625, if you do the multiplication).

Likewise, (x4)3 = x4 3 = x12.

This leads to another rule for exponents—the Power Rule for Exponents. To simplify a power of a power, you multiply the exponents, keeping the base the same. For example, (23)5 = 215.

 The Power Rule for Exponents For any positive number x and integers a and b: (xa)b= xa· b.

 Example Problem Simplify.6(c4)2 6(c4)2 Since you are raising a power to a power, apply the Power Rule and multiply exponents to simplify. The coefficient remains unchanged because it is outside of the parentheses. Answer 6(c4)2 =6c8

 Example Problem Simplify.a2(a5)3 Raise a5 to the power of 3 by multiplying the exponents together (the Power Rule). Since the exponents share the same base, a, they can be combined (the Product Rule). Answer Simplify: A) B) C) D) Show/Hide AnswerA) Incorrect. This expression is not simplified yet. Recall that –a can also be written –a1. Multiply –a1 by a8 to arrive at the correct answer. The correct answer is . B) Incorrect. Do not add the exponents of 2 and 4 together. The Power Rule states that for a power of a power you multiply the exponents. The correct answer is . C) Incorrect. Do not add the exponents of 2 and 4 together. The Power Rule states that for a power of a power you multiply the exponents. The correct answer is . D) Correct. Using the Power Rule, .

The Quotient Rule for Exponents

Let’s look at dividing terms containing exponential expressions. What happens if you divide two numbers in exponential form with the same base? Consider the following expression. You can rewrite the expression as: . Then you can cancel the common factors of 4 in the numerator and denominator: Finally, this expression can be rewritten as 43 using exponential notation. Notice that the exponent, 3, is the difference between the two exponents in the original expression, 5 and 2.

So, = 45-2 = 43.

Be careful that you subtract the exponent in the denominator from the exponent in the numerator. or = x79 = x-2

So, to divide two exponential terms with the same base, subtract the exponents.

Notice that = 40. And we know that = = 1. So this may help to explain why 40 = 1.

 Example Problem Evaluate.  These two exponents have the same base, 4. According to the Quotient Rule, you can subtract the power in the denominator from the power in the numerator. = 45

When dividing terms that also contain coefficients, divide the coefficients and then divide variable powers with the same base by subtracting the exponents.

 Example Problem Simplify.  Separate into numerical and variable factors. Since the bases of the exponents are the same, you can apply the Quotient Rule. Divide the coefficients and subtract the exponents of matching variables. Answer = All of these rules of exponents—the Product Rule, the Power Rule, and the Quotient Rule—are helpful when evaluating expressions with common bases.

 Example Problem Evaluate when x = 4. Separate into numerical and variable factors. Divide coefficients, and subtract the exponents of the variables. Simplify. Substitute the value 4 for the variable x. Answer = 768

Usually, it is easier to simplify the expression before substituting any values for your variables, but you will get the same answer either way.

 Example Problem Simplify.  Use the order of operations with PEMDAS:E: Evaluate exponents. Use the Power Rule to simplify (a5)3.  M: Multiply, using the Product Rule as the bases are the same.  D: Divide using the Quotient Rule. = There are rules that help when multiplying and dividing exponential expressions with the same base. To multiply two exponential terms with the same base, add their exponents. To raise a power to a power, multiply the exponents. To divide two exponential terms with the same base, subtract the exponents. Sours: https://www.montereyinstitute.org/courses/DevelopmentalMath/COURSE_TEXT2_RESOURCE/U11_L1_T2_text_final.html

2.3: Evaluate, Simplify, and Translate Expressions (Part 1)

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Skills to Develop

• Evaluate algebraic expressions
• Identify terms, coefficients, and like terms
• Simplify expressions by combining like terms
• Translate word phrases to algebraic expressions

Be prepared!

Before you get started, take this readiness quiz.

1. Is $$n ÷ 5$$ an expression or an equation? If you missed this problem, review Example 2.1.4.
2. Simplify $$4^5$$. If you missed this problem, review Example 2.1.6.
3. Simplify $$1 + 8 • 9$$. If you missed this problem, review Example 2.1.8.

Evaluate Algebraic Expressions

In the last section, we simplified expressions using the order of operations. In this section, we’ll evaluate expressions—again following the order of operations.

To evaluate an algebraic expression means to find the value of the expression when the variable is replaced by a given number. To evaluate an expression, we substitute the given number for the variable in the expression and then simplify the expression using the order of operations.

Example $$\PageIndex{1}$$: evaluate

Evaluate $$x + 7$$ when

1. $$x = 3$$
2. $$x = 12$$

Solution

1. To evaluate, substitute $$3$$ for $$x$$ in the expression, and then simplify.
 $$x + 7$$ Substitute. $$\textcolor{red}{3} + 7$$ Add. $$10$$

When $$x = 3$$, the expression $$x + 7$$ has a value of $$10$$.

1. To evaluate, substitute $$12$$ for $$x$$ in the expression, and then simplify.
 $$x + 7$$ Substitute. $$\textcolor{red}{12} + 7$$ Add. $$19$$

When $$x = 12$$, the expression $$x + 7$$ has a value of $$19$$. Notice that we got different results for parts (a) and (b) even though we started with the same expression. This is because the values used for $$x$$ were different. When we evaluate an expression, the value varies depending on the value used for the variable.

exercise $$\PageIndex{1}$$

Evaluate: $$y + 4$$ when

1. $$y = 6$$
2. $$y = 15$$

$$10$$

$$19$$

exercise $$\PageIndex{2}$$

Evaluate: $$a − 5$$ when

1. $$a = 9$$
2. $$a = 17$$

$$4$$

$$12$$

Example $$\PageIndex{2}$$

Evaluate $$9x − 2$$, when

1. $$x = 5$$
2. $$x = 1$$

Solution

Remember $$ab$$ means $$a$$ times $$b$$, so $$9x$$ means $$9$$ times $$x$$.

1. To evaluate the expression when $$x = 5$$, we substitute $$5$$ for $$x$$, and then simplify.
 $$9x - 2$$ Substitute $$\textcolor{red}{5}$$ for x. $$9 \cdot \textcolor{red}{5} - 2$$ Multiply. $$45 - 2$$ Subtract. $$43$$
1. To evaluate the expression when $$x = 1$$, we substitute $$1$$ for $$x$$, and then simplify.
 $$9x - 2$$ Substitute $$\textcolor{red}{1}$$ for x. $$9 \cdot \textcolor{red}{1} - 2$$ Multiply. $$9 - 2$$ Subtract. $$7$$

Notice that in part (a) that we wrote $$9 • 5$$ and in part (b) we wrote $$9(1)$$. Both the dot and the parentheses tell us to multiply.

exercise $$\PageIndex{3}$$

Evaluate: $$8x − 3$$, when

1. $$x = 2$$
2. $$x = 1$$

$$13$$

$$5$$

exercise $$\PageIndex{4}$$

Evaluate: $$4y − 4$$, when

1. $$y = 3$$
2. $$y = 5$$

$$8$$

$$16$$

Example $$\PageIndex{3}$$: evaluate

Evaluate $$x^2$$ when $$x = 10$$.

Solution

We substitute $$10$$ for $$x$$, and then simplify the expression.

 $$x^{2}$$ Substitute $$\textcolor{red}{10}$$ for x. $$\textcolor{red}{10}^{2}$$ Use the definition of exponent. $$10 \cdot 10$$ Multiply $$100$$

When $$x = 10$$, the expression $$x^2$$has a value of $$100$$.

exercise $$\PageIndex{5}$$

Evaluate: $$x^2$$ when $$x = 8$$.

$$64$$

exercise $$\PageIndex{6}$$

Evaluate: $$x^3$$when $$x = 6$$.

$$216$$

Example $$\PageIndex{4}$$: evaluate

Evaluate $$2^x$$ when $$x = 5$$.

Solution

In this expression, the variable is an exponent.

 $$2^{x}$$ Substitute $$\textcolor{red}{5}$$ for x. $$2^{\textcolor{red}{5}}$$ Use the definition of exponent. $$2 \cdot 2 \cdot 2 \cdot 2 \cdot 2$$ Multiply $$32$$

When $$x = 5$$, the expression $$2^x$$ has a value of $$32$$.

exercise $$\PageIndex{7}$$

Evaluate: $$2^x$$ when $$x = 6$$.

$$64$$

exercise $$\PageIndex{8}$$

Evaluate: $$3^x$$ when $$x = 4$$.

$$81$$

Example $$\PageIndex{5}$$: evaluate

Evaluate $$3x + 4y − 6$$ when $$x = 10$$ and $$y = 2$$.

Solution

This expression contains two variables, so we must make two substitutions.

 $$3x + 4y − 6$$ Substitute $$\textcolor{red}{10}$$ for x and $$\textcolor{blue}{2}$$ for y. $$3(\textcolor{red}{10}) + 4(\textcolor{blue}{2}) − 6$$ Multiply. $$30 + 8 - 6$$ Add and subtract left to right. $$32$$

When $$x = 10$$ and $$y = 2$$, the expression $$3x + 4y − 6$$ has a value of $$32$$.

exercise $$\PageIndex{9}$$

Evaluate: $$2x + 5y − 4$$ when $$x = 11$$ and $$y = 3$$

$$33$$

exercise $$\PageIndex{10}$$

Evaluate: $$5x − 2y − 9$$ when $$x = 7$$ and $$y = 8$$

$$10$$

Example $$\PageIndex{6}$$: evaluate

Evaluate $$2x^2 + 3x + 8$$ when $$x = 4$$.

Solution

We need to be careful when an expression has a variable with an exponent. In this expression, $$2x^2$$ means $$2 • x • x$$ and is different from the expression $$(2x)^2$$, which means $$2x • 2x$$.

 $$2x^{2} + 3x + 8$$ Substitute $$\textcolor{red}{4}$$ for each x. $$2(\textcolor{red}{4})^{2} + 3(\textcolor{red}{4}) + 8$$ Simplify 42. $$2(16) + 3(4) + 8$$ Multiply. $$32 + 12 + 8$$ Add. $$52$$

exercise $$\PageIndex{11}$$

Evaluate: $$3x^2 + 4x + 1$$ when $$x = 3$$.

$$40$$

exercise $$\PageIndex{12}$$

Evaluate: $$6x^2 − 4x − 7$$ when $$x = 2$$.

$$9$$

Identify Terms, Coefficients, and Like Terms

Algebraic expressions are made up of terms. A term is a constant or the product of a constant and one or more variables. Some examples of terms are $$7$$, $$y$$, $$5x^2$$, $$9a$$, and $$13xy$$.

The constant that multiplies the variable(s) in a term is called the coefficient. We can think of the coefficient as the number in front of the variable. The coefficient of the term $$3x$$ is $$3$$. When we write $$x$$, the coefficient is $$1$$, since $$x = 1 • x$$. Table $$\PageIndex{1}$$ gives the coefficients for each of the terms in the left column.

TermCoefficient
77
9a9
y1
5x25

An algebraic expression may consist of one or more terms added or subtracted. In this chapter, we will only work with terms that are added together. Table $$\PageIndex{2}$$ gives some examples of algebraic expressions with various numbers of terms. Notice that we include the operation before a term with it.

ExpressionTerms
77
yy
x + 7x, 7
2x + 7y + 42x, 7y, 4
3x2 + 4x2 + 5y + 33x2, 4x2, 5y, 3

Example $$\PageIndex{7}$$:

Identify each term in the expression $$9b + 15x^2 + a + 6$$. Then identify the coefficient of each term.

Solution

The expression has four terms. They are $$9b$$, $$15x^2$$, $$a$$, and $$6$$.

The coefficient of $$9b$$ is $$9$$.

The coefficient of $$15x^2$$ is $$15$$.

Remember that if no number is written before a variable, the coefficient is $$1$$. So the coefficient of a is $$1$$.

The coefficient of a constant is the constant, so the coefficient of $$6$$ is $$6$$.

exercise $$\PageIndex{13}$$

Identify all terms in the given expression, and their coefficients: $$4x + 3b + 2$$

The terms are $$4x, 3b,$$ and $$2$$. The coefficients are $$4, 3,$$ and $$2$$.

exercise $$\PageIndex{14}$$

Identify all terms in the given expression, and their coefficients: $$9a + 13a^2 + a^3$$

The terms are $$9a, 13a^2,$$ and $$a^3$$, The coefficients are $$9, 13,$$ and $$1$$.

Some terms share common traits. Look at the following terms. Which ones seem to have traits in common?

$$5x, 7, n^{2}, 4, 3x, 9n^{2}$$

Which of these terms are like terms?

• The terms $$7$$ and $$4$$ are both constant terms.
• The terms $$5x$$ and $$3x$$ are both terms with $$x$$.
• The terms $$n^2$$ and $$9n^2$$ both have $$n^2$$.

Terms are called like terms if they have the same variables and exponents. All constant terms are also like terms. So among the terms $$5x, 7, n^2, 4, 3x, 9n^2, 7$$ and $$4$$ are like terms, $$5x$$ and $$3x$$ are like terms, and $$n^2$$ and $$9n^2$$ are like terms.

Definition: Like terms

Terms that are either constants or have the same variables with the same exponents are like terms.

Example $$\PageIndex{8}$$: identify

Identify the like terms:

1. $$y^3, 7x^2, 14, 23, 4y^3, 9x, 5x^2$$
2. $$4x^2 + 2x + 5x^2 + 6x + 40x + 8xy$$

Solution

1. $$y^3, 7x^2, 14, 23, 4y^3, 9x, 5x^2$$

Look at the variables and exponents. The expression contains $$y^3, x^2, x$$, and constants. The terms $$y^3$$ and $$4y^3$$ are like terms because they both have $$y^3$$. The terms $$7x^2$$ and $$5x^2$$ are like terms because they both have $$x^2$$. The terms $$14$$ and $$23$$ are like terms because they are both constants. The term $$9x$$ does not have any like terms in this list since no other terms have the variable $$x$$ raised to the power of $$1$$.

1. $$4x^2 + 2x + 5x^2 + 6x + 40x + 8xy$$

Look at the variables and exponents. The expression contains the terms $$4x^2, 2x, 5x^2, 6x, 40x$$, and $$8xy$$ The terms $$4x^2$$ and $$5x^2$$ are like terms because they both have $$x^2$$. The terms $$2x, 6x$$, and $$40x$$ are like terms because they all have $$x$$. The term $$8xy$$ has no like terms in the given expression because no other terms contain the two variables $$xy$$.

exercise $$\PageIndex{15}$$

Identify the like terms in the list or the expression: $$9, 2x^3, y^2, 8x^3, 15, 9y, 11y^2$$

$$9, 15$$; $$2x^3$$ and $$8x^3$$, $$y^2$$, and $$11y^2$$

exercise $$\PageIndex{16}$$

Identify the like terms in the list or the expression: $$4x^3 + 8x^2 + 19 + 3x^2 + 24 + 6x^3$$

$$4x^3$$ and $$6x^3$$; $$8x^2$$ and $$3x^2$$; $$19$$ and $$24$$

Simplify Expressions by Combining Like Terms

We can simplify an expression by combining the like terms. What do you think $$3x + 6x$$ would simplify to? If you thought $$9x$$, you would be right!

We can see why this works by writing both terms as addition problems. Add the coefficients and keep the same variable. It doesn’t matter what $$x$$ is. If you have $$3$$ of something and add $$6$$ more of the same thing, the result is $$9$$ of them. For example, $$3$$ oranges plus $$6$$ oranges is $$9$$ oranges. We will discuss the mathematical properties behind this later.

The expression $$3x + 6x$$ has only two terms. When an expression contains more terms, it may be helpful to rearrange the terms so that like terms are together. The Commutative Property of Addition says that we can change the order of addends without changing the sum. So we could rearrange the following expression before combining like terms. Now it is easier to see the like terms to be combined.

HOW TO: COMBINE LIKE TERMS

Step 1. Identify like terms.

Step 2. Rearrange the expression so like terms are together.

Step 3. Add the coefficients of the like terms.

Example $$\PageIndex{9}$$: simplify

Simplify the expression: $$3x + 7 + 4x + 5$$.

Solution

 $$3x + 7 + 4x + 5$$ Identify the like terms $$\textcolor{red}{3x} + \textcolor{blue}{7} + \textcolor{red}{4x} + \textcolor{blue}{5}$$ Rearrange the expression, so the like terms are together. $$\textcolor{red}{3x} + \textcolor{red}{4x} + \textcolor{blue}{7} + \textcolor{blue}{5}$$ Add the coefficients of the like terms. $$\textcolor{red}{7x} + \textcolor{blue}{12}$$ The original expression is simplified to... $$7x + 12$$

exercise $$\PageIndex{17}$$

Simplify: $$7x + 9 + 9x + 8$$

$$16x+17$$

exercise $$\PageIndex{18}$$

Simplify: $$5y + 2 + 8y + 4y + 5$$

$$17y+7$$

Example $$\PageIndex{10}$$: simplify

Simplify the expression: $$7x^2 + 8x + x^2 + 4x$$.

Solution

 $$7x^{2} + 8x + x^{2} + 4x$$ Identify the like terms. $$\textcolor{red}{7x^{2}} + \textcolor{blue}{8x} + \textcolor{red}{x^{2}} + \textcolor{blue}{4x}$$ Rearrange the expression so like terms are together. $$\textcolor{red}{7x^{2}} + \textcolor{red}{x^{2}} + \textcolor{blue}{8x} + \textcolor{blue}{4x}$$ Add the coefficients of the like terms. $$\textcolor{red}{8x^{2}} + \textcolor{blue}{12x}$$

These are not like terms and cannot be combined. So $$8x^2 + 12x$$ is in simplest form.

exercise $$\PageIndex{19}$$

Simplify: $$3x^2 + 9x + x^2 + 5x$$

$$4x^2+14x$$

exercise $$\PageIndex{20}$$

Simplify: $$11y^2 + 8y + y^2 + 7y$$

$$12y^2+15y$$

Sours: https://math.libretexts.org/Bookshelves/PreAlgebra/Book%3A_Prealgebra_(OpenStax)/02%3A_Introduction_to_the_Language_of_Algebra/2.03%3A_Evaluate_Simplify_and_Translate_Expressions_(Part_1)
How to simplify (x+5)^2

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Equation Solving

Solve equations in one or more variables both symbolically and numerically.

Solve a polynomial equation:

Solve a system of linear equations:

Solve an equation with parameters:

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Polynomials

Solve, plot and find alternate forms of polynomial expressions in one or more variables.

Compute properties of a polynomial in several variables:

Factor a polynomial:

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Rational Functions

Compute discontinuities and other properties of rational functions.

Compute properties of a rational function:

Compute a partial fraction decomposition:

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Simplification

Simplify algebraic functions and expressions.

Simplify an expression:

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Matrices

Find properties and perform computations on matrices.

Do basic arithmetic on matrices:

{{0,-1},{1,0}}.{{1,2},{3,4}}+{{2,-1},{-1,2}}

Compute eigenvalues and eigenvectors of a matrix:

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Quaternions

Perform computations with the quaternion number system.

Do calculations with quaternions:

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Finite Groups

Discover properties of groups containing a finite number of elements.

Get information about a finite group:

Do algebra with permutations:

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Finite Fields

Discover properties of fields containing a finite number of elements.

Compute properties of a finite field:

Compute a specific property:

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Domain & Range

Find the domain and range of mathematical functions.

Compute the domain of a function:

Compute the range of a function:

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Sours: https://www.wolframalpha.com/examples/mathematics/algebra/

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How to Simplify Exponents

Multiply exponents by adding the exponents together. For example, x to the fifth power multiplied by x to the fourtth power equals x to the ninth power (x5 + x4 = x9), or (xxxxx)(xxxx) = (xxxxxxxxx).

Divide exponents by subtracting the exponents from each other. The equation x to the ninth power divided by x to the fifth power simplifies to x to the fourth power (x9 – x5 = x4), or (xxxxxxxxx)/(xxxxx) = (xxxx).

Simplify an exponent raised to another power by multiplying the exponents together. Simplifying x to the third power raised to the fourth power produces x to the 12th power [(x3)4 = x12], or (xxx)(xxx)(xxx)(xxx) = (xxxxxxxxxxxx).

Remember that any number to the 0th power equals one, meaning x to any power raised to the 0th power simplifies to one. Examples include x0 = 1, (x4)0 = 1, and (x5y3)0 = 1.

Note that equations with different variables such as x squared multiplied by y cubed (x2y3) cannot be combined to produce xy to the sixth power. This equation is already simplified. However, if the entire equation of x squared multiplied by y cubed is then squared, each of the variables is simplified separately, resulting in x to the fourth power multiplied by y to the sixth power (x2y3)2 = x4y6, or (xxxx)(yyyyyy).

Sours: https://sciencing.com/simplify-exponents-7248304.html

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