You may have seen teeny small numbers floating above and to the appropriate of other numbers and variables, like in the expression x3. What is that tiny 3 doing increase there? Parasailing? Is it little because the scale, perhaps because it is actually as big as the earth"s sunlight in reality, but is only seen from numerous thousands of miles away? No, that little guy is referred to as the exponent or power of x in the expression, and indeed it has a very an effective role in algebra.

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Big points Come in small Packages

The duty of an exponent is to save you time and to clean increase the method expressions room written. Basically, one exponent is a shorthand method to suggest repeated multiplication.

In the language the algebra, x3 (read "x to the 3rd power") method "x multiply by itself 3 times", or x x x. To find the worth of genuine numbers elevated to exponents, simply multiply the large number attached to the exponent (called the base) by itself the indicated variety of times.

Example 2: advice the exponential expressions.

(a) 43Solution: In this expression, 4 is the base and also 3 is the exponent. To discover the answer, main point 4 by chin 3 times:4 4 4 = 16 4 = 64Therefore, 43 = 64.(b) (-2)5Solution: In this case, the basic is -2, therefore it have to be multiply by itself 5 times. Don"t tension out about the negative signs. Simply go left come right, and multiply 2 numbers at a time. Begin with (-2) (-2) to obtain 4 and then main point that result by the next -2, and also that an outcome by the next -2 till you"re finished.(-2)(-2)(-2)(-2)(-2) = 4(-2)(-2)(-2) = -8(-2)(-2) = 16(-2) = -32
Critical Point

Two exponents have special names. Anything elevated to the 2nd power is said to be squared (52 have the right to be check out "5 squared"), and also anything to the 3rd power is stated to be cubed (x3 deserve to be check out "x cubed").


Exponential Rules

Once you create something in exponential form, there space very details rules you need to follow to leveling expressions. Here are the five most important rules, each through a brief explanation:

Rule 1: xa xb = xa + b. If exponential expressions through the very same base are multiplied, the an outcome is the common base raised to the amount of the powers.
x4 x7 = x4 + 7 = x11(22)(23) = 22 + 3 = 25
Rule 2: xaxb = xa-b. If you are splitting exponential expressions through the same base, the result is the usual base raised to the distinction of the 2 powers.z7z4 = z7 - 4 = z3(-5)10(-5)9 = (-5)1 = -5
Critical Point

Any number increased to the 1 power equals the original number (x1 = x); so, if there"s no strength written, it"s construed to it is in 1 (7 = 71). In addition, anything (except 0) increased to the 0 power equals 1 (x0 = 1, 120 = 1). The expression 00 functions a small differently, yet you don"t address that till calculus.


Rule 3: (xa)b = xa b. If an exponential expression is itself increased to a power, main point the exponents together. This is various from dominance 1, since here there is one base elevated to 2 powers, and also in dominion 1, over there were two bases increased to 2 powers.(35)6 = 35 6 = 330 (k2)0 = k2 0 = k0 = 1Rule 4: (xy)a = xayaand (xy)a = xaya. If a product (multiplication problem) or quotient (division problem) or any type of such combination is elevated to a power, then so is every individual piece within.(5y)2 = 52 y2 = 25y2 (x2y3)4 = (x2)4 (y3)4 = x8y12Rule 5: x-a = 1x-aand 1x-a = xa. If other is elevated to a negative power, relocate it to the other component of the fraction (if it"s in the numerator, send it come the denominator and vice versa) and readjust the exponent to its opposite. If the expression consists of other confident exponents, leaving them alone.Most teachers think about answers containing an adverse exponents unsimplified, for this reason make certain to eliminate an adverse exponents native your last answer. Also, keep in mind that elevating something to the -1 power is tantamount to taking its reciprocal.
x-3y2z3 = y2x3z3(42w5)-1 = 42(-1)w5(-1) = 4-2w-5 = w516

Most that the time, you"ll have actually to use multiple rules throughout the same problem, in her attempts to simplify.

Example 3: simplify the expression (x2y-3)2(xy2)4.

Solution: begin by using Rules 3 and also 4 to the numerator and denominator.

x2 2y-3 2x1 4y2 4 = x4y-6x4y8

Now use Rule 2, since you have matching bases in the numerator and also denominator.

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Excerpted indigenous The finish Idiot"s guide to Algebra 2004 through W. Michael Kelley. All legal rights reserved including the best of reproduction in entirety or in component in any type of form. Used by setup with Alpha Books, a member that Penguin team (USA) Inc.