1.2 Sub Chapter Notes

Unit: Exponents and Polynomials

Module: Understanding Exponents

An Introduction to Exponents

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Exponents: A shortcut way to write multiplication problems in which a number multiplies with itself. The exponent tells how many times the number is multiplying with itself. Powers: Another name for exponents.

Exponents are written as smaller numbers to the right and above (superscripted) the number or variable which they have counted.

Exponents tell you how many of the number to which they are attached are being multiplied together.

Exponents are sometimes called “powers.”

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Unit: Exponents and Polynomials

Module: Understanding Exponents

Evaluating Exponential Expressions

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Exponents: The superscripted (or small up in the air) numbers which often are attached to the right side of a number or variable. They indicate how many times that number or variable is multiplied with itself. Ex., 24 = 2 · 2 · 2 · 2. Exponentiation: The notation of using the superscripted numbers instead of writing a number out however many times as a multiplication problem. Order of Operations: When evaluating an expression that contains more than one operation, first check to see what must be done within parentheses. Perform the work required by exponents next. Then do multiplication and division working from left to right. Finally, finish up with the addition and subtraction, again working from left to right.

Exponents tell you how many times a number is multiplying with itself.

Note: Watch signs carefully. In this example, you cannot assume that you are squaring –3. You must assume that the negative is separate from the 3 so that you will square and have a negative answer, -9.

Does –x2 mean (-x)2 or –(x2)? It means –(x2). The negative is added after squaring unless it is included in parentheses.

The order in which you do arithmetic matters. 1. Always do the multiplication indicated by exponents first. 2. Then, do the regular multiplication and division. 3. Finally, do the adding and subtracting.

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6357 –rev 04/19/2001

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Unit: Exponents and Polynomials

Module: Understanding Exponents

[Page 1 of 1]

Applying the Rules of Exponents

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Multiplying with Exponents: Add the exponents attached to the same bases. You’re still just counting how many times a particular number is going to multiply with itself in a particular problem. Base: The number or variable to which an exponent is attached. Grouping with Exponents: All the bases that carry the same exponent can be written together in parenthesis with the exponent written once and attached to the parenthesis. Ungrouping with Exponents: You can ungroup bases enclosed in parenthesis with an exponent attached if you carefully apply the exponent to each base as you remove the parenthesis. Multiply the exponents for individual terms with the one for the whole group so that you maintain an accurate count of how many times each base number is used in the problem.

Multiplying with Exponents: Add exponents attached to the same bases. You’re not yet doing any multiplication to get an answer; you’re still just counting how many numbers are going to multiply. Remember: The exponent is just a shortcut for how many times a number is multiplying with itself during a multiplication problem. Multiplying with Exponents: In this example, you cannot add any exponents because they are attached to different bases. Since the two bases are to the same power, you can group and multiply the bases instead. Grouping numbers for multiplication or division is allowed if they carry the same exponent.

To ungroup numbers, apply the exponent attached to the parenthesis to each factor that is within the parenthesis. Note: The outside exponent will multiply with each inside exponent because now you are counting for each factor in the group.

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6531 –rev 04/23/2001

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Unit: Exponents and Polynomials

Module: Understanding Exponents

[Page 1 of 1]

Evaluating Expressions with Negative Exponents

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To Divide with Exponents: Subtract the exponent for any factor in the denominator from the exponent for the same factor in the numerator to get the exponent for that factor in the answer. 0 as an Exponent: Any number to the 0 power equals 1; i.e. any a 0 = 1. Negative Exponents: The negative tells you that the base to which the exponent is attached belongs in the opposite level of the fraction. When you move the factor to the other level, the exponent becomes positive.

Dividing with exponents is a matter of subtracting the top exponent by the bottom (or dividing) exponent. What you are actually doing is canceling a number from the top and the bottom with each subtraction; i.e., you are reducing the fraction. 0 as an exponent. Any number raised to the 0 power, i.e., any a 0, equals 1. You can think about this fact by considering the fraction a1/a1. On one hand, the numerator and denominator of this fraction cancel, so the whole fraction is equal to 1. On the other hand, by subtracting exponents, this fraction is equal to a1-1=a0. Thus you can see that a0 = 1. This is true for every nonzero number a. Negative Exponents: A negative exponent attached to a base tells you that the base belongs in the opposite level of the fraction. Move it there and the exponent becomes positive. In this example, the exponent –2 becomes +2 when the base, (-3), moves from the numerator to the denominator. If you wish to move a number out of the denominator, use the same process. Change the sign of its exponent and move it to the numerator.

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6533 –rev 04/23/2001

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