Xto The Power Of 3

Behold the Power of Exponents

You may accept seen teeny piddling numbers floating above and to the correct of other numbers and variables, like in the expression ten ³. What is that little 3 doing upwards in that location? Parasailing? Is information technology small because of scale, mayhap because it is actually equally large every bit the globe'southward sun in reality, but is only seen from hundreds of thousands of miles away? No, that little guy is chosen the exponent or ability of x in the expression, and indeed it has a very powerful part in algebra.

Talk the Talk

In the exponential expression y ⁴, iv is the exponent and y is the base.

Large Things Come up in Pocket-size Packages

The role of an exponent is to save you lot time and to clean upward the way expressions are written. Basically, an exponent is a shorthand way to indicate repeated multiplication.

In the language of algebra, ten3 (read "ten to the tertiary power") ways "x multiplied by itself three times", or x · x · x. To discover the value of existent numbers raised to exponents, just multiply the large number attached to the exponent (called the base) by itself the indicated number of times.

Instance 2: Evaluate the exponential expressions.

(a) 4³
Solution: In this expression, 4 is the base and 3 is the exponent. To find the answer, multiply 4 by itself 3 times:
4 · 4 · 4 = xvi · iv = 64
Therefore, 4ⁱⁱⁱ = 64.
(b) (-2)⁵
Solution: In this instance, the base is -2, so information technology should be multiplied past itself v times. Don't stress out about the negative signs. Simply go left to right, and multiply two numbers at a fourth dimension. Start with (-2) · (-2) to become 4 and and so multiply that result by the side by side -2, and that result by the next -2 until you're finished.
(-2)(-2)(-2)(-two)(-2) = 4(-ii)(-2)(-ii) = -8(-2)(-2) = sixteen(-2) = -32

Disquisitional Point

2 exponents have special names. Anything raised to the second power is said to exist squared (5² can be read "five squared"), and anything to the third power is said to be cubed (x³ can be read "x cubed").

You've Got Problems

Problem 2: Evaluate the expression: (-iii)⁴.

Exponential Rules

In one case you write something in exponential form, there are very specific rules you lot must follow to simplify expressions. Here are the 5 almost of import rules, each with a brief explanation:

Rule ane: x ^a · 10 ^b = x ^{a + b}. If exponential expressions with the same base are multiplied, the result is the common base raised to the sum of the powers.

10 ^four · x ⁷ = x ^{4 + 7} = ten ¹¹

(2²)(2³) = two^{two + 3} = two⁵

Rule 2: ^{10 ^a} _{10 ^b} = x ^a-b. If you are dividing exponential expressions with the same base, the result is the mutual base raised to the difference of the ii powers.

^{z ⁷} _{z ⁴} = z ^{vii - 4} = z^three ^(-5)¹⁰ _(-v)⁹ = (-v)^one = -5

Critical Bespeak

Any number raised to the one power equals the original number (ten ¹ = x); so, if there'south no power written, information technology's understood to exist 1 (7 = 7^one). In add-on, anything (except 0) raised to the 0 power equals 1 (x⁰ = ane, 12⁰ = 1). The expression 0⁰ works a little differently, simply you don't deal with that until calculus.

Rule 3: (x ^a)^b = x ^{a · b}. If an exponential expression is itself raised to a power, multiply the exponents together. This is dissimilar from Rule one, because here there is one base raised to ii powers, and in Rule 1, at that place were 2 bases raised to ii powers.

(3^v)^half-dozen = iii^{5 · half dozen} = three³⁰ (k ⁱⁱ)⁰ = k ^{2 · 0} = thousand ⁰ = 1

Rule 4: ( xy )^a = x ^a y ^a and (^x _y)^a = ^ten ^a _{y ^a}. If a product (multiplication problem) or quotient (division trouble) or any such combination is raised to a power, then so is every private piece within.

(5y)² = v² · y ² = 25y ² (x²y³)^iv = (x ²)^iv · (y ³)⁴ = x ⁸ y ¹²

Rule five: x ^-a = ^ane _{x ^{- a}} and ^one _{x ^-a} = 10 ^a. If something is raised to a negative power, motion it to the other part of the fraction (if it's in the numerator, transport it to the denominator and vice versa) and change the exponent to its opposite. If the expression contains other positive exponents, leave them solitary.

Almost teachers consider answers containing negative exponents unsimplified, then make sure to eliminate negative exponents from your final answer. Also, annotation that raising something to the -one power is equivalent to taking its reciprocal.

^{10 ^-3 y ²} _{z ³} = ^{y ²} _{ten ⁱⁱⁱ z³}

(^{4^two} _{w ⁵})^-ane = ^{4^2(-1)} _{west ^v(-1)} = ^{iv^-2} _{due west ^-five} = ^w⁵ ₁₆

Nigh of the time, y'all'll have to apply multiple rules during the same problem, in your attempts to simplify.

Example iii: Simplify the expression (^{x ⁱⁱ y ^-3)²} _{(xy ²)^iv}.

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

^{x ^{two · 2} y ^{-three · ii}} _{x ^{1 · 4y ^{2 · iv}}} = ^{10 ⁴ y ^-6} _{x ⁴ y ⁸}

Now apply Dominion 2, since yous accept matching bases in the numerator and denominator.

Y'all've Got Bug

Problem 3: Simplify the expression (x ^threey)⁵ · (x ^-2y²)^three.

(ten ^{4 - four})(y ^{-6 - 8}) = x ⁰ y ^-xiv = y ^-xiv

Finish by applying Dominion 5.

y ^-14 = ¹ _{y ¹⁴}

CIG Algebra

Excerpted from The Complete Idiot's Guide to Algebra © 2004 by W. Michael Kelley. All rights reserved including the correct of reproduction in whole or in part in any form. Used by organisation with Alpha Books, a member of Penguin Group (United states) Inc.

You can purchase this volume at Amazon.com and Barnes & Noble.