Overview: Exponents and Variables
An exponent is a signal for how many times to multiply a number over again. For example, 26 means the same thing as 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 = 64. If a variable has an exponent, for example, a2, the exponent 2 just means a ∙ a. When evaluating variables, it is important to understand the meaning of exponents and how they relate in the number sentence.
Bases and Exponents
A number such as 26 or x12 has two parts, the base, and the exponent. The 2 is the base, and it is raised to the sixth power. Similarly, the x is the base, and it is raised to the 12th power. The bases must be the same in order to combine exponents, and to reduce the monomial to its simplest form. Therefore, a monomial such as 2c3x2 is already factored as much as it can be and is in its simplest form.
Multiplying and Dividing Exponents
If a monomial is 2c3c2, it is not factored into its simplest form, but is in two parts with the same base that can be combined, 2c3 and c2. They can be multiplied to form a new monomial, 2c3 ∙ c ∙ c , or 2c5, by a property of exponents that states that for any real integer x, when a and b are positive integers, xa∙ xb = xa +b. That is one can multiply exponents with the same base by adding exponents. Dividing exponents works similarly, by subtracting exponents when they are the same bases. Therefore , for any real number x, when x is not equal to 0, and a and b are positive integers, xa/xb = xa-b. The exponent in the denominator is always subtracted from the exponent in the numerator.
Negative Exponents
Negative exponents follow the logic of dividing exponents of like bases. By applying the property for dividing exponents (The Quotient Property), x 8/x10, would equal x8-10, or x-2. What is x-2? Take a step back to solve the division problem another way. Using this method, x8/x10 can be expressed as x8/x8∙1/x2. Therefore, x-2 means the same thing as 1/x2. Generalizing to another definition, for any real number x, when x is not equal to 0, and a is a positive integer, x-a means the same thing as i/xa.
Scientific Notation
Scientific notation is an application of the rules about exponents, and are a way to express very large numbers (such as the distances to planets, stars, and galaxies) and very small numbers (such as the size of microbes, viruses, and atoms. When the number is large, the power of 10 will have a positive exponent, but when it is small, the power of 10 will have a negative exponent. For example, if the nearest star outside the solar system is about 4 X 1013 km from the earth, it is 40,000,000,000,000 km from the earth. Similarly, a helium atom has a diameter of 2.2 x 10-8 cm, or 0.00000022 cm.
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