One number is 5 times another. Write an expression for the sum of their reciprocals. Then simplify the expression. Use x for the variable. Evaluate the expression for the given value of the variable. Check for extraneous solutions. I really need help with both of these. The same rule applies to expressions with variables.
The following statements therefore hold true:. Each of the other rules for operating on numbers applies to expressions with variables as well. You will see how each of these applies in the following examples.
To simplify the first part of the expression, apply the rule for multiplying two exponential expressions with the same base:. To simplify the second part of the expression, apply the rule for multiplying numbers with exponents:. We can also apply the rule for raising a power to another exponent:. The addition and subtraction of rational expressions are bound by all of the same rules as the addition and subtraction of fractions.
Adding and subtracting fractions should be a familiar process, and we will rely on this concept in our discussion of adding and subtracting rational expressions. The key is finding the least common denominator of the two rational expressions: the smallest multiple of both denominators. Then, you rewrite the two fractions using this denominator.
Finally, you add or subtract the fractions by combining the numerators and leaving the denominator alone. You could probably find the least common denominator if you played around with the numbers long enough.
Here, we will show you a systematic method for finding least common denominators—a method that works with rational expressions just as well as it does with numbers. We start, as usual, by factoring. For each of the denominators, we find all the prime factors —i. If you are not familiar with the concept of prime factors, it may take a few minutes to get used to. Similarly, the prime factors of 30 are 2, 3, and 5. But why does this help? Similarly, any number whose prime factors include a 2, a 3, and a 5 will be a multiple of Prime Factors of Fractions: Finding the prime factors of the denominators of two fractions enables us to find a common denominator.
The least common denominator is the smallest number that contains the overlap of both factored denominators: in this case, it must have two 2s, one 3, and one 5. This may look like a very strange way of solving problems that you have known how to solve since the third grade. However, you should spend a few minutes carefully following the above solution.
When applying this strategy to rational expressions, first look at the denominators of the two rational expressions and see if they are the same. If they are the same, then simply add or subtract the numerators from each other, leaving the denominator alone. If the two denominators are different, however, then you will need to use the above strategy of finding the least common denominator. When we add or subtract rational expressions, we will not simply be considering the prime factors of integers when looking for the least common denominator.
Rather, we will be looking for monomial and binomial factors that are common to both rational expressions. This requires factoring algebraic expressions. We begin problems of this type by factoring.
Notice that we can rewrite the first denominator in terms of its factors. The following are some key words and phrases and their translations:. Addition: sum, plus, add to, more than, increased by, total. Subtraction: difference of, minus, subtracted from, less than, decreased by, less. Multiplication: product, times, multiply, twice, of. Division: quotient divide, into, ratio. Example Write the phrase as an algebraic expression. In this example, we are not evaluating an expression, so we will not be coming up with a value.
However, we are wanting to rewrite it as an algebraic expression. Again, we are wanting to rewrite this as an algebraic expression, not evaluate it.
Since an equation is two expressions set equal to each other, we will be using the same mathematical translations we did above. The difference is we will have an equal sign between the two expressions. Example Write the sentence as an equation. Do you remember what quotient translates into?
If you said division , you are doing great. Do you remember what less than translates into? If you said subtraction , you are doing great. These are practice problems to help bring you to the next level.
It will allow you to check and see if you have an understanding of these types of problems. Math works just like anything else, if you want to get good at it, then you need to practice it. Even the best athletes and musicians had help along the way and lots of practice, practice, practice, to get good at their sport or instrument. In fact there is no such thing as too much practice. The product of 12 and a number. The following are webpages that can assist you in the topics that were covered on this page:.
Exponential Notation. In this problem, what is the base? If you said 5, you are correct! What is the exponent? If you said 4, you are right! If you said 7, you are correct! Reducing or Simplifying Fractions. Factor the top and bottom until you get factors that cannot be factored further. If you find the same factor on both the top and bottom, you can cancel them.
If after factoring the top and bottom as much as possible, if there are no common factors in the top and bottom, the fraction is reduced to lowest terms. Adding Subtracting Fractions.
If you have common denominators, add subtract the numerators. If not, find common denominators. To find common denominators, factor all the denominators and fill in the missing factors. You can multiply the bottom by whatever you want so long as you multiply the top by the same thing. Multiply the tops and multiply the bottoms. You can cancel either before or after you multiply. Invert the divisor and multiply.
You can make any change you want on one side of an equation so long as you make the same change on the other side. One of the most common techniques is to get rid of a term on one side by subtracting it from both sides.
When you get rid of a term on one side, it pops up on the other side with its sign changed. Moving a term from one side to the other and changing its sign is called transposing the term. If you move a factor from one side to the other, move it across the fraction bar.
Steps in solving first degree equations. Clear Denominators: Multiply both sides by a common denominator.
Simplify: Remove parentheses and combine like terms. Transpose known terms to one side and unknown terms to the other. Divide both sides by the coefficient of the unknown.
Steps in solving quadratic equations by factoring. Steps in solving quadratic equations by completing the square. Steps in solving quadratic equations using the quadratic formula.
Clear denominators: Multiply both sides by a common denominator. Transpose all terms to one side leaving a 0 on the other. Substitute the coefficients into the quadratic formula. Polynomial equations of degree higher than two are beyond the scope of this discussion. There is a cubic and a quartic formula, involving radicals, to get rid of the powers, but beyond that it can be proven that solution by radicals will not always work.
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