SKILL IN ALGEBRA (ALGEBRIC EXPRESSIONS)
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The four operations and their signs
The four basic mathematical operations are
Addition (+): The process of combining two or more numbers or quantities to find their sum.
Subtraction (-): The process of finding the difference between two numbers or quantities. It is the inverse operation of addition.
Multiplication (*): The process of repeated addition of a number or quantity. It is the process of finding the product of two or more numbers or quantities.
Division (/): The process of finding how many times one number or quantity is contained within another. It is the inverse operation of multiplication.
The signs used to represent these operations in mathematical notation are + for addition, - for subtraction, * for multiplication and / for division.
The function of parentheses
Parentheses, also known as round brackets, are used in mathematics and programming to group together expressions or operations that need to be performed before others. They indicate that the operations inside the parentheses should be done first, before any other operations in the same equation or statement.
For example, in the equation (2 + 3) * 4, the operations inside the parentheses (2 + 3) should be done first, resulting in 5. Then, the result of (2 + 3) is multiplied by 4, resulting in 20. Without the parentheses, the equation would be 2 + 3 * 4, which would result in 14 because the multiplication would be done before the addition.
In programming, parentheses are used to indicate the order of operations and to group together function arguments or elements in an array.
Parentheses also used to set apart expressions that are not to be read or executed as part of the main statement or command. This can be helpful to make the code more readable and to prevent misinterpretation.
Terms versus factors
In mathematics, terms and factors are related concepts but have different meanings.
A term is a numerical or algebraic expression, such as a constant, variable, or the product of constants and variables. For example, in the algebraic expression 3x^2 + 5x - 2, 3x^2, 5x and -2 are terms.
Factors, on the other hand, are the numbers or expressions that are multiplied together to form a product. For example, in the expression 3x^2, 3 and x^2 are factors. Factors can also refer to the integers that divide into a given number without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4 and 12.
In summary, a term is a single element in an algebraic expression or equation, while a factor is one of the components that make up a product or a number.
Powers and exponents
In mathematics, a power or exponent is a way to indicate how many times a number, called the base, is multiplied by itself. The exponent is the number written above and to the right of the base, and it tells you how many times the base is being used as a factor.
For example, in the expression 2^3, 2 is the base and 3 is the exponent. This expression can be read as "two raised to the power of three" or "two to the third power" and it means 2 multiplied by itself 3 times, which is equal to 8 (2 x 2 x 2 = 8).
Another example is 5^2, 5 is the base and 2 is the exponent, this expression can be read as "five raised to the power of two" or "five squared" and it means 5 multiplied by itself 2 times, which is 25 (5 x 5 = 25)
Exponents can also be written using a superscript notation, like 2^3=2³
It's also important to mention that any number raised to the power of 1 is equal to itself, and any number raised to the power of 0 is equal to 1, except for 0^0 which is not defined.
A negative exponent means to take the reciprocal of the base raised to the absolute value of the exponent.
The order of operations
The order of operations is a set of rules that dictate the order in which mathematical operations should be performed in order to correctly solve an equation or expression. The acronym PEMDAS is often used to help remember the order of operations:
P: Parentheses - Perform any calculations inside parentheses first.
E: Exponents - Perform any exponents (ie powers or roots).
MD: Multiplication and Division - Perform all multiplications and divisions, from left to right.
AS: Addition and Subtraction - Perform all additions and subtractions, from left to right.
It is important to follow the order of operations because if you don't, you may get a different answer. For example, if you are solving the expression 4 + 2 x 3, if you perform the multiplication first, you will get the wrong answer of 6. If you follow the order of operations and first, the Parentheses, Exponents, Multiplication, Division, Addition and Subtraction, you will get the correct answer of 10.
Following this order of operations ensures that the mathematical expressions are evaluated consistently and correctly every time.
Values and evaluations
In mathematics, a value refers to the numerical or algebraic result of an expression or equation. For example, the value of the expression 2+3 is 5, and the value of the equation x+2=5 when x=3 is 3.
Evaluation is the process of finding the value of an expression or equation by substituting numbers for variables and performing the necessary mathematical operations according to the order of operations.
For example, to evaluate the expression 2x+3 for x=4, the value of x, which is 4, is substituted for x, and the expression becomes 2(4) + 3 = 8+3 = 11.
Evaluation can also be applied to algebraic equations, when we try to find the value of a variable. For example, to evaluate the equation x+2=5, we can solve for x by subtracting 2 from both sides and get x=3.
When evaluating expressions or equations, it's important to use the correct order of operations, and to substitute the values into the expressions or equations correctly.
It's also important to mention that the evaluation doesn't always have to be numerical, it can also be symbolic, where the result is an algebraic expression.
Evaluating algebraic expressions
Evaluating algebraic expressions involves substituting a specific value for each variable in the expression and then performing the necessary mathematical operations according to the order of operations.
For example, to evaluate the expression 2x+3 for x=4, the value of x, which is 4, is substituted for x, and the expression becomes 2(4) + 3 = 8+3 = 11.
Another example, to evaluate the expression 3x^2-5x+2 for x=2, the value of x, which is 2, is substituted for x, and the expression becomes 3(2^2) - 5(2) + 2 = 3(4) - 10 + 2 = 12 - 10 + 2 = 2
It's important to remember to use the order of operations when evaluating algebraic expressions. Parentheses must be done first, then exponents, then multiplication and division, and finally addition and subtraction.
When evaluating an expression, you can also simplify it, in order to make it more readable and easy to understand. You can do this by combining like terms, or using the distributive property of multiplication over addition.
It's important to mention that it's possible to evaluate algebraic expressions symbolically, which means that the result is another algebraic expression, and not a numerical value
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