Simplifying Exponential Expressions
Simplifying exponential expressions can be a little bit tricky at first but I am here to help clear up how to deal with them. When you multiply like bases, like x^2*x^3, you would ADD together the exponents to get the answer of x^5. If they weren't like bases, you wouldn't be allowed to do this. Keep in mind that you don't add the two x's together, just the exponents. When you divide like bases, like x^8/x^2, you would subtract the ^2 from the top and bottom of the fraction, leaving you with x^6. Or, if you had x^2/x^8, you could subtract the ^8 from the top and bottom of the fraction to get x^-6 or you could subtract the ^2 from the top and the bottom of the fraction to get 1/x^6. When raising a power to a power, like (x^2)^3, you would MULTIPLY the exponents to get the answer of x^6. When raising a product to a power, such as (xy)^2, you would do what the exponent says to do, so in this case we would square xy to get (x*y)(x*y), and then multiplied together, the answer would be x^2*y^2 . When raising a quotient to a power, like (x/y)^2, you would begin by applying the exponent to the thing that is in parentheses, which would be (x/y)(x/y), and then simplify, and multiply to get the final answer, x^2/y^2. To end my discussion on simplifying exponential expressions, anything to the zeroeth power is 1 and anything to the ^-1 would be the reciprocal of itself, for example x^-1 would be 1/x.
Closed Sets
A closed set is a set of numbers where if an operation is performed on members of the set that they will still produce a member of the same set. For example, all even numbers will always add up to even numbers, which makes even numbers a closed set under addition. However, odd numbers will never add up to odd number; they will always add up to an even number, so odd numbers are not a closed set under addition.
Multiplying Polynomial Expressions
Let's say you have an expression like (2x+4)(3x+2) and you need to solve it. This kind of expression is called a polynomial expression, and I am going to teach you how to solve it. The basis of solving polynomial expressions is all about using the distributive property. So, we will multiply 2x times 3x and 2, and then multiply 4 times 3x and 2, which will equal 6x^2+4x+12x+8 which simplifies to 6x^2+16x+8, which is the final answer. I hope this helped you figure out how to multiply polynomial expressions!
Solving Absolute Value Equations
Absolute value is the distance a number is from zero. An absolute value equation would look something like this: |x+3|=9. To solve this, you will have to set the equation equal to 9 and then -9 WITHOUT the absolute value bars and find out both possible solutions. We will begin by setting the equation equal to 9, which is x+3=9. You would then simply subtract 3 from both sides of the equation, which leaves you with one answer, x=6. Now, we will set the equation equal to -9, which is x+3=-9. You would now subtract 3 form both sides of the equation, which leaves you with x=-12, the other possible answer. You can then test each solution in the equation and make sure you did your math right! I hope my explanation helps you with solving absolute value equations!
Simplifying exponential expressions can be a little bit tricky at first but I am here to help clear up how to deal with them. When you multiply like bases, like x^2*x^3, you would ADD together the exponents to get the answer of x^5. If they weren't like bases, you wouldn't be allowed to do this. Keep in mind that you don't add the two x's together, just the exponents. When you divide like bases, like x^8/x^2, you would subtract the ^2 from the top and bottom of the fraction, leaving you with x^6. Or, if you had x^2/x^8, you could subtract the ^8 from the top and bottom of the fraction to get x^-6 or you could subtract the ^2 from the top and the bottom of the fraction to get 1/x^6. When raising a power to a power, like (x^2)^3, you would MULTIPLY the exponents to get the answer of x^6. When raising a product to a power, such as (xy)^2, you would do what the exponent says to do, so in this case we would square xy to get (x*y)(x*y), and then multiplied together, the answer would be x^2*y^2 . When raising a quotient to a power, like (x/y)^2, you would begin by applying the exponent to the thing that is in parentheses, which would be (x/y)(x/y), and then simplify, and multiply to get the final answer, x^2/y^2. To end my discussion on simplifying exponential expressions, anything to the zeroeth power is 1 and anything to the ^-1 would be the reciprocal of itself, for example x^-1 would be 1/x.
Closed Sets
A closed set is a set of numbers where if an operation is performed on members of the set that they will still produce a member of the same set. For example, all even numbers will always add up to even numbers, which makes even numbers a closed set under addition. However, odd numbers will never add up to odd number; they will always add up to an even number, so odd numbers are not a closed set under addition.
Multiplying Polynomial Expressions
Let's say you have an expression like (2x+4)(3x+2) and you need to solve it. This kind of expression is called a polynomial expression, and I am going to teach you how to solve it. The basis of solving polynomial expressions is all about using the distributive property. So, we will multiply 2x times 3x and 2, and then multiply 4 times 3x and 2, which will equal 6x^2+4x+12x+8 which simplifies to 6x^2+16x+8, which is the final answer. I hope this helped you figure out how to multiply polynomial expressions!
Solving Absolute Value Equations
Absolute value is the distance a number is from zero. An absolute value equation would look something like this: |x+3|=9. To solve this, you will have to set the equation equal to 9 and then -9 WITHOUT the absolute value bars and find out both possible solutions. We will begin by setting the equation equal to 9, which is x+3=9. You would then simply subtract 3 from both sides of the equation, which leaves you with one answer, x=6. Now, we will set the equation equal to -9, which is x+3=-9. You would now subtract 3 form both sides of the equation, which leaves you with x=-12, the other possible answer. You can then test each solution in the equation and make sure you did your math right! I hope my explanation helps you with solving absolute value equations!
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