You may have actually seen teeny small numbers floating above and to the best of other numbers and also variables, prefer in the expression x3. What is that tiny 3 doing up there? Parasailing? Is it small because of scale, perhaps since it is actually as large as the earth"s sunlight in reality, but is only seen from thousands of thousands of mile away? No, that little guy is referred to as the exponent or strength of x in the expression, and also indeed it has a very powerful role in algebra.

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Big points Come in tiny Packages

The duty of an exponent is to conserve you time and also to clean up the means expressions space written. Basically, one exponent is a shorthand means to suggest repeated multiplication.

In the language of algebra, x3 (read "x come the third power") means "x multiplied by itself three times", or x x x. To discover the value of real numbers increased to exponents, simply multiply the large number attached to the exponent (called the base) by itself the indicated variety of times.

Example 2: evaluate the exponential expressions.

(a) 43Solution: In this expression, 4 is the base and 3 is the exponent. To find the answer, multiply 4 by chin 3 times:4 4 4 = 16 4 = 64Therefore, 43 = 64.(b) (-2)5Solution: In this case, the base is -2, so it must be multiply by chin 5 times. Don"t stress and anxiety out about the an unfavorable signs. Just go left come right, and multiply 2 numbers at a time. Begin with (-2) (-2) to get 4 and then main point that an outcome by the next -2, and that result by the following -2 until you"re finished.(-2)(-2)(-2)(-2)(-2) = 4(-2)(-2)(-2) = -8(-2)(-2) = 16(-2) = -32
Critical Point

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


Exponential Rules

Once you compose something in exponential form, there room very certain rules you need to follow to simplify expressions. Below are the 5 most important rules, each through a short explanation:

Rule 1: xa xb = xa + b. If exponential expressions v the same base room multiplied, the result is the usual base increased to the sum of the powers.
x4 x7 = x4 + 7 = x11(22)(23) = 22 + 3 = 25
Rule 2: xaxb = xa-b. If friend are separating exponential expressions v the exact same base, the result is the common base elevated to the distinction of the 2 powers.z7z4 = z7 - 4 = z3(-5)10(-5)9 = (-5)1 = -5
Critical Point

Any number elevated to the 1 power equates to the original number (x1 = x); so, if there"s no power written, it"s understood to it is in 1 (7 = 71). In addition, anything (except 0) elevated 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 one exponential expression is itself raised to a power, main point the exponents together. This is different from preeminence 1, due to the fact that here there is one base elevated to two powers, and in dominion 1, over there were 2 bases raised 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 such mix is increased to a power, then so is every individual item within.(5y)2 = 52 y2 = 25y2 (x2y3)4 = (x2)4 (y3)4 = x8y12Rule 5: x-a = 1x-aand 1x-a = xa. If miscellaneous is elevated to a negative power, relocate it come the other part 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 includes other confident exponents, leave them alone.Most teachers take into consideration answers containing an unfavorable exponents unsimplified, for this reason make certain to eliminate an unfavorable exponents indigenous your last answer. Also, keep in mind that raising something to the -1 power is equivalent to acquisition its reciprocal.
x-3y2z3 = y2x3z3(42w5)-1 = 42(-1)w5(-1) = 4-2w-5 = w516

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

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

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

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

Now apply Rule 2, because you have equivalent bases in the numerator and also denominator.

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Excerpted native The complete Idiot"s overview to Algebra 2004 by W. Michael Kelley. All civil liberties reserved including the appropriate of reproduction in entirety or in part in any kind of form. Offered by arrangement with Alpha Books, a member the Penguin team (USA) Inc.