Exponents. Common Powers

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1 Exponents An exponent defines the number of times a number is to be multiplied by itself. For example, in a b, where a is the base and b the exponent, a is multiplied by itself btimes. In a numerical example, 2 5 = An exponent can also be referred to as a power: a number with an exponent of 2 is raised to the second power. The following are other terms related to exponents with which you should be familiar: Base. The base refers to the 3 in 3 5. It is the number that is being multiplied by itself however many times specified by the exponent. Exponent. The exponent is the 5 in 3 5. It indicates the number of times the base is to be multiplied with itself. Square. Saying that a number is squared means that the number has been raised to the second power, i.e., that it has an exponent of 2. In the expression 6 2, 6 has been squared. Cube. Saying that a number is cubed means that it has been raised to the third power, i.e., that it has an exponent of 3. In the expression 4 3, 4 has been cubed. Common Powers It may be worthwhile to memorize a few common powers before taking the Math IIC, in order to save the time you d take to calculate them during the test. Here is a list of squares from 1 through 10. Memorizing the first few cubes might also be helpful: 1/5

2 The first few powers of 2 are also useful to know for many applications: Adding and Subtracting Numbers with Exponents In order to add or subtract numbers with exponents, you must first find the value of each power, then add the two numbers. For example, to add , you must expand the exponents to get (3 3 3) + (4 4), and then, = 43. However, algebraic expressions that have the same bases and exponents, such as 3x 4 and 5x 4, can be added and subtracted. For example, 3x 4 + 5x 4 = 8x 4. Multiplying and Dividing Numbers with Exponents To multiply exponential numbers raised to the same exponent, raise their product to that To divide exponential numbers raised to the same exponent, raise their quotient to that 2/5

3 To multiply exponential numbers or tems that have the same base, add the exponents together: To divide two same-base exponential numbers or terms, just subtract the exponents: If you need to multiply or divide two exponential numbers that don t have the same base or exponent, you ll just have to do your work the old-fashioned way: multiply the exponential numbers out, and multiply or divide them accordingly. Raising an Exponent to an Exponent Occasionally you might encounter a power raised to another power, as in (3 2 ) 4 and (x 4 ) 3. In such cases, multiply the exponents: Exponents and Fractions To raise a fraction to an exponent, raise both the numerator and the denominator to that Exponents and Negative Numbers As we said in the negative numbers section, when you multiply a negative number by a negative number, you get a positive number, and when you multiply a negative number by a positive number, you get a negative number. These rules affect how negative numbers function in reference to exponents. 3/5

4 When you raise a negative number to an even number exponent, you get a positive number. For example, ( 2) 4 = 16. To see why this is so, let s break down the example. ( 2) 4 means When you multiply the first two 2s together, you get 4 because you are multiplying two negative numbers. Then, when you multiply the 4 by the next 2, you get 8, since you are multiplying a positive number by a negative number. Finally, you multiply the 8 by the last 2 and get 16, since you re once again multiplying two negative numbers. When you raise a negative number to an odd power, you get a negative number. To see why, refer to the example above and stop the process at 8, which equals ( 2) 3. These rules can help a great deal as you go about eliminating answer choices and checking potential correct answers. For example, if you have a negative number raised to an odd power and you get a positive answer, you know your answer is wrong. Likewise, on that same question, you could eliminate any answer choices that are positive. Special Exponents There are a few special properties of certain exponents that you also should know. Zero Any base raised to the power of zero is equal to 1. If you see any exponent of the formx 0, you should know that its value is 1. Note, however, that 0 0 is undefined. One Any base raised to the power of one is equal to itself. For example, 2 1 = 2, ( 67) 1 = 67, and x 1 = x. This can be helpful when you re attempting an operation on exponential terms with the same base. For example: Fractional Exponents Exponents can be fractions, too. When a number or term is raised to a fractional power, it is called taking the root of that number or term. This expression can be converted into a more convenient form: For example, is equal to the fifth root of 2 to the thirteenth power: 4/5

5 The symbol is also known as the radical, and anything under the radical (in this case 2 13 ) is called the radicand. For a more familiar example, look at 9 1 2, which is the same as : Negative Exponents Seeing a negative number as a power may be a little strange the first time around. But the principle at work is simple. Any number or term raised to a negative power is equal to the reciprocal of that base raised to the opposite power. For example: Or a slightly more complicated example: You ve got the four rules of special exponents. Here are some examples to firm up your knowledge: 5/5