![]() In general if n is a positive integer, then factoring 10^n from a decimal number moves the decimal point n places to the left, and factoring 10^-n moves the decimal point n places to the right. We see that factoring 10^5 from 132,000 moved the decimal point five places to the left, while factoring 10^-5 from 0.0000132 moved the decimal point five places to the right. Integral exponents can be used to write such numbers in a compact form. Large and small numbers in decimal form occur in scientific work. Click on "Solve Similar" button to see more examples. Let’s see how our math solver solves this and similar problems. We use the above definitions to simplify each of the following.Įxample 4. Furthermore, we will rewrite them without zero or negative exponents.Įxample 1. In the following examples we will use the new definitions and laws of exponents to simplify the given expressions. In fact E.5 may be written in the simpler form It is straightforward but tedious to show that all the laws of Section 5.1 hold for integral exponents. Note that in the above definitions, when the exponent is zero or negative, a cannot be zero. This definition gives us the following helpful rule Since 1/a^n is the only real number such that (1/a^n)a^n=1. Similarly, if n is a positive integer, then in order For a^-n to satisfy E.1, We would have Since 1 is the only real number such that 1a^n=a^n, we define If a!=0, then in order for a^0 to satisfy E.1, we would have We want laws E.1 through E.5 to hold for this larger set of exponents. In this section we will enlarge our set of exponents to include zero and the negative integers. ![]() Let’s see how our math solver simplifies this and similar problems. We use the laws of exponents to compute each of the following.Įxample 3. Each of these factors has n factors of x, so that altogether there are nm factors of x. In E.3 the expression (x^n)^m has m factors of x^n. Using associativity and commutativity of multiplication we have Since (xy)^n has n factors of xy, there are n factors of x and n factors of y. We will establish E.2 and E.3 below and leave E.4 and the rest of E.5 as exercises for the interested reader. Law E.1 and the first two parts of E.5 were established in Chapter 2. The justification of these laws involves nothing more than counting the number of factors in a given expression. If x and y are real numbers and m and n are positive integers, then Computations with exponents depend on the following five basic laws. The number x is called the base and n the exponent or power. Where x is any real number and n is a positive integer. Any radical expression can be written with a rational exponent, which we call exponential form An equivalent expression written using a rational exponent.In Chapter 3 we introduced the notation that is equivalent to a radical where the denominator is the index and the numerator is the exponent. In general, given any nonzero real number a where m and n are positive integers ( n ≥ 2),Īn expression with a rational exponent The fractional exponent m/ n that indicates a radical with index n and exponent m: a m / n = a m n. In other words, 5 2 / 3 is a cube root of 5 2 and we can write: ![]() This shows that 5 2 / 3 is one of three equal factors of 5 2. Next, consider fractional exponents where the numerator is an integer other than 1. In other words, the denominator of a fractional exponent determines the index of an nth root. This is true in general, given any nonzero real number a and integer n ≥ 2, Therefore, 2 1 / 3 is a cube root of 2, and we can write Here 5 1 / 2 is one of two equal factors of 5 hence it is a square root of 5, and we can writeįurthermore, we can see that 2 1 / 3 is one of three equal factors of 2.Ģ 1 / 3 ⋅ 2 1 / 3 ⋅ 2 1 / 3 = 2 1 / 3 + 1 / 3 + 1 / 3 = 2 3 / 3 = 2 1 = 2 Given any rational numbers m and n, we haveįor example, if we have an exponent of 1/2, then the product rule for exponents implies the following:ĥ 1 / 2 ⋅ 5 1 / 2 = 5 1 / 2 + 1 / 2 = 5 1 = 5 In particular, recall the product rule for exponents. All of the rules for exponents developed up to this point apply. ![]() In this section, we will define what rational (or fractional) exponents mean and how to work with them. So far, exponents have been limited to integers. ![]()
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