Reduction of equations online. How to simplify an algebraic expression
The exponent is used to make it easier to write the operation of multiplying a number by itself. For example, instead of writing, you can write 4 5 (\displaystyle 4^(5))(an explanation of such a transition is given in the first section of this article). Powers make it easier to write long or complex expressions or equations; also, powers are easily added and subtracted, resulting in a simplification of an expression or equation (for example, 4 2 ∗ 4 3 = 4 5 (\displaystyle 4^(2)*4^(3)=4^(5))).
Note: if you need to decide exponential equation(in such an equation the unknown is in the exponent), read .
Steps
Solving simple problems with powers
- 4 5 = 4 ∗ 4 ∗ 4 ∗ 4 ∗ 4 (\displaystyle 4^(5)=4*4*4*4*4)
- 4 ∗ 4 = 16 (\displaystyle 4*4=16)
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Multiply the result (16 in our example) by the next number. Each subsequent result will increase proportionally. In our example, multiply 16 by 4. Like this:
- 4 5 = 16 ∗ 4 ∗ 4 ∗ 4 (\displaystyle 4^(5)=16*4*4*4)
- 16 ∗ 4 = 64 (\displaystyle 16*4=64)
- 4 5 = 64 ∗ 4 ∗ 4 (\displaystyle 4^(5)=64*4*4)
- 64 ∗ 4 = 256 (\displaystyle 64*4=256)
- 4 5 = 256 ∗ 4 (\displaystyle 4^(5)=256*4)
- 256 ∗ 4 = 1024 (\displaystyle 256*4=1024)
- Keep multiplying the result of multiplying the first two numbers by the next number until you get the final answer. To do this, multiply the first two numbers, and then multiply the result by the next number in the sequence. This method is valid for any degree. In our example, you should get: 4 5 = 4 ∗ 4 ∗ 4 ∗ 4 ∗ 4 = 1024 (\displaystyle 4^(5)=4*4*4*4*4=1024) .
- 4 5 = 16 ∗ 4 ∗ 4 ∗ 4 (\displaystyle 4^(5)=16*4*4*4)
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Solve the following problems. Check your answer with a calculator.
- 8 2 (\displaystyle 8^(2))
- 3 4 (\displaystyle 3^(4))
- 10 7 (\displaystyle 10^(7))
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On the calculator, look for the key labeled "exp", or " x n (\displaystyle x^(n))", or "^". With this key you will raise a number to a power. It is practically impossible to manually calculate the degree with a large exponent (for example, the degree 9 15 (\displaystyle 9^(15))), but the calculator can easily cope with this task. In Windows 7, the standard calculator can be switched to engineering mode; to do this, click "View" -\u003e "Engineering". To switch to normal mode, click "View" -\u003e "Normal".
- Check the received answer using a search engine (Google or Yandex). Using the "^" key on the computer keyboard, enter the expression into the search engine, which will instantly display the correct answer (and possibly suggest similar expressions for study).
Addition, subtraction, multiplication of powers
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You can add and subtract powers only if they have the same base. If you need to add powers with the same bases and exponents, then you can replace the addition operation with a multiplication operation. For example, given the expression 4 5 + 4 5 (\displaystyle 4^(5)+4^(5)). Remember that the degree 4 5 (\displaystyle 4^(5)) can be represented as 1 ∗ 4 5 (\displaystyle 1*4^(5)); thus, 4 5 + 4 5 = 1 ∗ 4 5 + 1 ∗ 4 5 = 2 ∗ 4 5 (\displaystyle 4^(5)+4^(5)=1*4^(5)+1*4^(5) =2*4^(5))(where 1 +1 =2). That is, count the number of similar degrees, and then multiply such a degree and this number. In our example, raise 4 to the fifth power, and then multiply the result by 2. Remember that the addition operation can be replaced by a multiplication operation, for example, 3 + 3 = 2 ∗ 3 (\displaystyle 3+3=2*3). Here are other examples:
- 3 2 + 3 2 = 2 ∗ 3 2 (\displaystyle 3^(2)+3^(2)=2*3^(2))
- 4 5 + 4 5 + 4 5 = 3 ∗ 4 5 (\displaystyle 4^(5)+4^(5)+4^(5)=3*4^(5))
- 4 5 − 4 5 + 2 = 2 (\displaystyle 4^(5)-4^(5)+2=2)
- 4 x 2 − 2 x 2 = 2 x 2 (\displaystyle 4x^(2)-2x^(2)=2x^(2))
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When multiplying powers with the same base their exponents are added (the base does not change). For example, given the expression x 2 ∗ x 5 (\displaystyle x^(2)*x^(5)). In this case, you just need to add the indicators, leaving the base unchanged. Thus, x 2 ∗ x 5 = x 7 (\displaystyle x^(2)*x^(5)=x^(7)). Here is a visual explanation of this rule:
When raising a power to a power, the exponents are multiplied. For example, given a degree. Since the exponents are multiplied, then (x 2) 5 = x 2 ∗ 5 = x 10 (\displaystyle (x^(2))^(5)=x^(2*5)=x^(10)). The meaning of this rule is that you multiply the power (x 2) (\displaystyle (x^(2))) on itself five times. Like this:
- (x 2) 5 (\displaystyle (x^(2))^(5))
- (x 2) 5 = x 2 ∗ x 2 ∗ x 2 ∗ x 2 ∗ x 2 (\displaystyle (x^(2))^(5)=x^(2)*x^(2)*x^( 2)*x^(2)*x^(2))
- Since the base is the same, the exponents simply add up: (x 2) 5 = x 2 ∗ x 2 ∗ x 2 ∗ x 2 ∗ x 2 = x 10 (\displaystyle (x^(2))^(5)=x^(2)*x^(2)* x^(2)*x^(2)*x^(2)=x^(10))
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An exponent with a negative exponent should be converted to a fraction (to the inverse power). It doesn't matter if you don't know what a reciprocal is. If you are given a degree with a negative exponent, for example, 3 − 2 (\displaystyle 3^(-2)), write this power in the denominator of the fraction (put 1 in the numerator), and make the exponent positive. In our example: 1 3 2 (\displaystyle (\frac (1)(3^(2)))). Here are other examples:
When dividing powers with the same base, their exponents are subtracted (the base does not change). The division operation is the opposite of the multiplication operation. For example, given the expression 4 4 4 2 (\displaystyle (\frac (4^(4))(4^(2)))). Subtract the exponent in the denominator from the exponent in the numerator (do not change the base). Thus, 4 4 4 2 = 4 4 − 2 = 4 2 (\displaystyle (\frac (4^(4))(4^(2)))=4^(4-2)=4^(2)) = 16 .
- The degree in the denominator can be written as follows: 1 4 2 (\displaystyle (\frac (1)(4^(2)))) = 4 − 2 (\displaystyle 4^(-2)). Remember that a fraction is a number (power, expression) with a negative exponent.
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Below are some expressions to help you learn how to solve power problems. The above expressions cover the material presented in this section. To see the answer, just highlight the empty space after the equals sign.
Solving problems with fractional exponents
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A degree with a fractional exponent (for example, ) is converted to a root extraction operation. In our example: x 1 2 (\displaystyle x^(\frac (1)(2))) = x(\displaystyle(\sqrt(x))). It does not matter what number is in the denominator of the fractional exponent. For example, x 1 4 (\displaystyle x^(\frac (1)(4))) is the fourth root of "x" x 4 (\displaystyle (\sqrt[(4)](x))) .
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If the exponent is an improper fraction, then such an exponent can be decomposed into two powers to simplify the solution of the problem. There is nothing complicated about this - just remember the rule for multiplying powers. For example, given a degree. Turn that exponent into a root whose exponent is equal to the denominator of the fractional exponent, and then raise that root to the exponent equal to the numerator of the fractional exponent. To do this, remember that 5 3 (\displaystyle (\frac (5)(3))) = (1 3) ∗ 5 (\displaystyle ((\frac (1)(3)))*5). In our example:
- x 5 3 (\displaystyle x^(\frac (5)(3)))
- x 1 3 = x 3 (\displaystyle x^(\frac (1)(3))=(\sqrt[(3)](x)))
- x 5 3 = x 5 ∗ x 1 3 (\displaystyle x^(\frac (5)(3))=x^(5)*x^(\frac (1)(3))) = (x 3) 5 (\displaystyle ((\sqrt[(3)](x)))^(5))
- Some calculators have a button for calculating exponents (first you need to enter the base, then press the button, and then enter the exponent). It is denoted as ^ or x^y.
- Remember that any number is equal to itself to the first power, for example, 4 1 = 4. (\displaystyle 4^(1)=4.) Moreover, any number multiplied or divided by one is equal to itself, for example, 5 ∗ 1 = 5 (\displaystyle 5*1=5) and 5 / 1 = 5 (\displaystyle 5/1=5).
- Know that the degree 0 0 does not exist (such a degree has no solution). When you try to solve such a degree on a calculator or on a computer, you will get an error. But remember that any number to the power of zero is equal to 1, for example, 4 0 = 1. (\displaystyle 4^(0)=1.)
- AT higher mathematics, which operates on imaginary numbers: e a i x = c o s a x + i s i n a x (\displaystyle e^(a)ix=cosax+isinax), where i = (− 1) (\displaystyle i=(\sqrt (())-1)); e is a constant approximately equal to 2.7; a is an arbitrary constant. The proof of this equality can be found in any textbook on higher mathematics.
Warnings
- As the exponent increases, its value greatly increases. Therefore, if the answer seems wrong to you, in fact it may turn out to be true. You can check this by plotting any exponential function, such as 2 x .
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Multiply the base of the exponent by itself a number of times equal to the exponent. If you need to solve a problem with exponents manually, rewrite the exponent as a multiplication operation, where the base of the exponent is multiplied by itself. For example, given the degree 3 4 (\displaystyle 3^(4)). In this case, the base of degree 3 must be multiplied by itself 4 times: 3 ∗ 3 ∗ 3 ∗ 3 (\displaystyle 3*3*3*3). Here are other examples:
First, multiply the first two numbers. For example, 4 5 (\displaystyle 4^(5)) = 4 ∗ 4 ∗ 4 ∗ 4 ∗ 4 (\displaystyle 4*4*4*4*4). Don't worry - the calculation process is not as complicated as it seems at first glance. First multiply the first two quadruples, and then replace them with the result. Like this:
§ 1 The concept of simplifying a literal expression
In this lesson, we will get acquainted with the concept of “similar terms” and, using examples, we will learn how to carry out the reduction of similar terms, thus simplifying literal expressions.
Let's find out the meaning of the concept of "simplification". The word "simplification" is derived from the word "simplify". To simplify means to make simple, simpler. Therefore, to simplify a literal expression is to make it shorter, with a minimum number of actions.
Consider the expression 9x + 4x. This is a literal expression that is a sum. The terms here are presented as products of a number and a letter. The numerical factor of such terms is called the coefficient. In this expression, the coefficients will be the numbers 9 and 4. Please note that the multiplier represented by the letter is the same in both terms of this sum.
Recall the distributive law of multiplication:
To multiply the sum by a number, you can multiply each term by this number and add the resulting products.
AT general view is written as follows: (a + b) ∙ c \u003d ac + bc.
This law is valid in both directions ac + bc = (a + b) ∙ c
Let's apply it to our literal expression: the sum of the products of 9x and 4x is equal to the product, the first factor of which is the sum of 9 and 4, the second factor is x.
9 + 4 = 13 makes 13x.
9x + 4x = (9 + 4)x = 13x.
Instead of three actions in the expression, one action remained - multiplication. So, we have made our literal expression simpler, i.e. simplified it.
§ 2 Reduction of like terms
The terms 9x and 4x differ only in their coefficients - such terms are called similar. The letter part of similar terms is the same. Similar terms also include numbers and equal terms.
For example, in the expression 9a + 12 - 15, the numbers 12 and -15 will be similar terms, and in the sum of the products of 12 and 6a, the numbers 14 and the products of 12 and 6a (12 ∙ 6a + 14 + 12 ∙ 6a), the equal terms represented by the product of 12 and 6a.
It is important to note that terms with equal coefficients and different literal factors are not similar, although it is sometimes useful to apply the distributive law of multiplication to them, for example, the sum of the products of 5x and 5y is equal to the product of the number 5 and the sum of x and y
5x + 5y = 5(x + y).
Let's simplify the expression -9a + 15a - 4 + 10.
In this case, the terms -9a and 15a are similar terms, since they differ only in their coefficients. They have the same letter multiplier, and the terms -4 and 10 are also similar, since they are numbers. We add like terms:
9a + 15a - 4 + 10
9a + 15a = 6a;
We get: 6a + 6.
Simplifying the expression, we found the sums of like terms, in mathematics this is called the reduction of like terms.
If bringing such terms is difficult, you can come up with words for them and add objects.
For example, consider the expression:
For each letter we take our own object: b-apple, c-pear, then it will turn out: 2 apples minus 5 pears plus 8 pears.
Can we subtract pears from apples? Of course not. But we can add 8 pears to minus 5 pears.
We give like terms -5 pears + 8 pears. Like terms have the same literal part, therefore, when reducing like terms, it is enough to add the coefficients and add the literal part to the result:
(-5 + 8) pears - you get 3 pears.
Returning to our literal expression, we have -5s + 8s = 3s. Thus, after reducing similar terms, we obtain the expression 2b + 3c.
So, in this lesson, you got acquainted with the concept of “similar terms” and learned how to simplify literal expressions by bringing like terms.
List of used literature:
- Mathematics. 6th grade: lesson plans to the textbook by I.I. Zubareva, A.G. Mordkovich // author-compiler L.A. Topilin. Mnemosyne 2009.
- Mathematics. Grade 6: student textbook educational institutions. I.I. Zubareva, A.G. Mordkovich.- M.: Mnemozina, 2013.
- Mathematics. Grade 6: textbook for educational institutions / G.V. Dorofeev, I.F. Sharygin, S.B. Suvorov and others / edited by G.V. Dorofeeva, I.F. Sharygin; Russian Academy of Sciences, Russian Academy of Education. M.: "Enlightenment", 2010.
- Mathematics. Grade 6: textbook for general educational institutions / N.Ya. Vilenkin, V.I. Zhokhov, A.S. Chesnokov, S.I. Schwarzburd. – M.: Mnemozina, 2013.
- Mathematics. Grade 6: textbook / G.K. Muravin, O.V. Ant. – M.: Bustard, 2014.
Used images:
Convenient and simple online calculator fractions with detailed solution maybe:
- Add, subtract, multiply and divide fractions online,
- Receive turnkey solution fractions with a picture and it is convenient to transfer it.
The result of solving fractions will be here ...
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Fraction sign "/" + - * :
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Our online fraction calculator has fast input. To get the solution of fractions, for example, just write 1/2+2/7
into the calculator and press the " solve fractions". The calculator will write you detailed solution of fractions and issue copy-friendly image.
The characters used for writing in the calculator
You can type an example for a solution both from the keyboard and using the buttons.![](https://i2.wp.com/reshit.ru/Servisi_dlya_uchashihsya/kalkulyator_drobey/img_servisa/onlain-kalkulyator-drobey.jpg)
Features of the online fraction calculator
The fraction calculator can only perform operations with 2 simple fractions. They can be either correct (the numerator is less than the denominator) or incorrect (the numerator is greater than the denominator). The numbers in the numerator and denominators cannot be negative and greater than 999.Our online calculator solves fractions and brings the answer to correct form- reduces the fraction and highlights the whole part, if necessary.
If you need to solve negative fractions, just use the minus properties. When multiplying and dividing negative fractions, minus by minus gives plus. That is, the product and division of negative fractions is equal to the product and division of the same positive ones. If one fraction is negative when multiplied or divided, then simply remove the minus, and then add it to the answer. When adding negative fractions, the result will be the same as if you added the same positive fractions. If you add one negative fraction, then this is the same as subtracting the same positive one.
When subtracting negative fractions, the result will be the same as if they were reversed and made positive. That is, a minus by a minus in this case gives a plus, and the sum does not change from a rearrangement of the terms. We use the same rules when subtracting fractions, one of which is negative.
To solve mixed fractions (fractions in which the whole part is highlighted), simply drive the whole part into a fraction. To do this, multiply the integer part by the denominator and add to the numerator.
If you need to solve 3 or more fractions online, then you should solve them one by one. First, count the first 2 fractions, then solve the next fraction with the answer received, and so on. Perform operations in turn for 2 fractions, and in the end you will get the correct answer.
Simplifying algebraic expressions is one of the key points learning algebra and an extremely useful skill for all mathematicians. Simplification allows you to reduce a complex or long expression to a simple expression that is easy to work with. Basic simplification skills are good even for those who are not enthusiastic about mathematics. Keeping a few simple rules, you can simplify many of the most common types of algebraic expressions without any special mathematical knowledge.
Steps
Important definitions
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Similar members. These are members with a variable of the same order, members with the same variables, or free members (members that do not contain a variable). In other words, like terms include one variable to the same extent, include several identical variables, or do not include a variable at all. The order of the terms in the expression does not matter.
- For example, 3x 2 and 4x 2 are like terms because they contain the variable "x" of the second order (in the second power). However, x and x 2 are not similar members, since they contain the variable "x" of different orders (first and second). Similarly, -3yx and 5xz are not similar members because they contain different variables.
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Factorization. This is finding such numbers, the product of which leads to the original number. Any original number can have several factors. For example, the number 12 can be decomposed into the following series of factors: 1 × 12, 2 × 6 and 3 × 4, so we can say that the numbers 1, 2, 3, 4, 6 and 12 are factors of the number 12. The factors are the same as divisors , that is, the numbers by which the original number is divisible.
- For example, if you want to factor the number 20, write it like this: 4×5.
- Note that when factoring, the variable is taken into account. For example, 20x = 4(5x).
- Prime numbers cannot be factored because they are only divisible by themselves and 1.
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Remember and follow the order of operations to avoid mistakes.
- Parentheses
- Degree
- Multiplication
- Division
- Addition
- Subtraction
Casting Like Members
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Write down the expression. The simplest algebraic expressions (which do not contain fractions, roots, and so on) can be solved (simplified) in just a few steps.
- For example, simplify the expression 1 + 2x - 3 + 4x.
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Define similar members (members with a variable of the same order, members with the same variables, or free members).
- Find similar terms in this expression. The terms 2x and 4x contain a variable of the same order (first). Also, 1 and -3 are free members (do not contain a variable). Thus, in this expression, the terms 2x and 4x are similar, and the members 1 and -3 are also similar.
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Give similar terms. This means adding or subtracting them and simplifying the expression.
- 2x+4x= 6x
- 1 - 3 = -2
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Rewrite the expression taking into account the given terms. You will get a simple expression with fewer terms. The new expression is equal to the original.
- In our example: 1 + 2x - 3 + 4x = 6x - 2, that is, the original expression is simplified and easier to work with.
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Observe the order in which operations are performed when casting like terms. In our example, it was easy to bring similar terms. However, in the case of complex expressions in which members are enclosed in brackets and fractions and roots are present, it is not so easy to bring such terms. In these cases, follow the order of operations.
- For example, consider the expression 5(3x - 1) + x((2x)/(2)) + 8 - 3x. Here it would be a mistake to immediately define 3x and 2x as like terms and quote them, because first you need to expand the parentheses. Therefore, perform the operations in their order.
- 5(3x-1) + x((2x)/(2)) + 8 - 3x
- 15x - 5 + x(x) + 8 - 3x
- 15x - 5 + x 2 + 8 - 3x. Now, when the expression contains only addition and subtraction operations, you can cast like terms.
- x 2 + (15x - 3x) + (8 - 5)
- x 2 + 12x + 3
- For example, consider the expression 5(3x - 1) + x((2x)/(2)) + 8 - 3x. Here it would be a mistake to immediately define 3x and 2x as like terms and quote them, because first you need to expand the parentheses. Therefore, perform the operations in their order.
Parenthesizing the multiplier
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Find the greatest common divisor (gcd) of all coefficients of the expression. NOD is largest number, by which all the coefficients of the expression are divided.
- For example, consider the equation 9x 2 + 27x - 3. In this case, gcd=3, since any coefficient of this expression is divisible by 3.
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Divide each term of the expression by gcd. The resulting terms will contain smaller coefficients than in the original expression.
- In our example, divide each expression term by 3.
- 9x2/3=3x2
- 27x/3=9x
- -3/3 = -1
- It turned out the expression 3x2 + 9x-1. It is not equal to the original expression.
- In our example, divide each expression term by 3.
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Write the original expression as equal to the product of gcd times the resulting expression. That is, enclose the resulting expression in brackets, and put the GCD out of brackets.
- In our example: 9x 2 + 27x - 3 = 3(3x 2 + 9x - 1)
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Simplifying fractional expressions by taking the multiplier out of brackets. Why just take the multiplier out of brackets, as was done earlier? Then, to learn how to simplify complex expressions, such as fractional expressions. In this case, parenthesizing the multiplier can help get rid of the fraction (the denominator).
- For example, consider fractional expression(9x 2 + 27x - 3)/3. Use parentheses to simplify this expression.
- Factor out the factor 3 (as you did before): (3(3x 2 + 9x - 1))/3
- Note that both the numerator and denominator now have the number 3. This can be reduced, and you get the expression: (3x 2 + 9x - 1) / 1
- Since any fraction that has the number 1 in the denominator is just equal to the numerator, the original fractional expression is simplified to: 3x2 + 9x-1.
- For example, consider fractional expression(9x 2 + 27x - 3)/3. Use parentheses to simplify this expression.
Additional Simplification Techniques
- Consider a simple example: √(90). The number 90 can be decomposed into the following factors: 9 and 10, and from 9 extract Square root(3) and take out 3 from under the root.
- √(90)
- √(9×10)
- √(9)×√(10)
- 3×√(10)
- 3√(10)
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Simplifying expressions with powers. In some expressions, there are operations of multiplication or division of terms with a degree. In the case of multiplication of terms with one base, their degrees are added; in the case of dividing terms with the same base, their degrees are subtracted.
- For example, consider the expression 6x 3 × 8x 4 + (x 17 / x 15). In the case of multiplication, add the exponents, and in the case of division, subtract them.
- 6x 3 × 8x 4 + (x 17 / x 15)
- (6 × 8)x 3 + 4 + (x 17 - 15)
- 48x7+x2
- The following is an explanation of the rule for multiplying and dividing terms with a degree.
- Multiplying terms with powers is equivalent to multiplying terms by themselves. For example, since x 3 = x × x × x and x 5 = x × x × x × x × x, then x 3 × x 5 = (x × x × x) × (x × x × x × x × x), or x 8 .
- Similarly, dividing terms with powers is equivalent to dividing terms by themselves. x 5 /x 3 \u003d (x × x × x × x × x) / (x × x × x). Since similar terms that are in both the numerator and the denominator can be reduced, the product of two "x", or x 2, remains in the numerator.
- For example, consider the expression 6x 3 × 8x 4 + (x 17 / x 15). In the case of multiplication, add the exponents, and in the case of division, subtract them.
- Always be aware of the signs (plus or minus) in front of the terms of an expression, as many people have difficulty choosing the right sign.
- Ask for help if needed!
- Simplifying algebraic expressions is not easy, but if you get your hands on it, you can use this skill for a lifetime.