Properties of Exponents Learn to apply the properties

The factors of a power, such as 74, can be grouped in different ways.

MULTIPLYING POWERS WITH THE SAME BASE Words Numbers To multiply powers with 35 •

Multiplying Powers with the Same Base Multiply. Write the product as one power. A.

Multiplying Powers with the Same Base Continued Multiply. Write the product as one power.

Notice what occurs when you divide powers with the same base. 5 5 55

Dividing Powers with the Same Base Divide. Write the quotient as one power. A.

Try This: Divide. Write the quotient as one power. A. 99 92 99 –

When the numerator and denominator have the same base and exponent, subtracting the exponents

Helpful Hint 00 does not exist because 00 represents a quotient of the form

THE ZERO POWER Words Numbers The zero power of any number except 0 equals

Practice Write the product or quotient as one power 1. n 3 n 4

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Properties of Exponents Learn to apply the properties of exponents and to evaluate the zero exponent.

The factors of a power, such as 74, can be grouped in different ways. Notice the relationship of the exponents in each product. 7 • 7 • 7 = 74 (7 • 7) • 7 = 73 • 71 = 74 (7 • 7) • (7 • 7) = 72 • 72 = 74

MULTIPLYING POWERS WITH THE SAME BASE Words Numbers To multiply powers with 35 • 3 8 = the same base, 5 + 8 3 = 313 keep the base and add the exponents. Algebra bm • bn = bm + n

Multiplying Powers with the Same Base Multiply. Write the product as one power. A. 66 • 63 66 + 3 69 Add exponents. B. n 5 • n 7 n 5 + 7 n 12 Add exponents.

Multiplying Powers with the Same Base Continued Multiply. Write the product as one power. C. 25 • 2 25 + 1 26 Think: 2 = 2 1 Add exponents. D. 244 • 244 24 4 + 4 24 8 Add exponents.

Notice what occurs when you divide powers with the same base. 5 5 55 2 = 5 • 5 = = = 5 5 5 53 DIVIDING POWERS WITH THE SAME BASE Words To divide powers with the same base, keep the base and subtract the exponents. Numbers 6 9 = 69 – 4 = 6 5 64 Algebra b m = bm – n bn

Dividing Powers with the Same Base Divide. Write the quotient as one power. A. 5 7 3 7 75 – 3 7 Subtract exponents. 2 10 B. x 9 x x 10 – 9 x Subtract exponents. Think: x 1 = x

Try This: Divide. Write the quotient as one power. A. 99 92 99 – 2 97 B. Subtract exponents. e 10 e 5 e 10 – 5 5 e Subtract exponents.

When the numerator and denominator have the same base and exponent, subtracting the exponents results in a 0 exponent. 2 4 2 – 2 = 40 = 1= 4 42 This result can be confirmed by writing out the factors. 42 42 (4 • 4) = 1 =1 = = (4 • 4) 1 (4 • 4)

Helpful Hint 00 does not exist because 00 represents a quotient of the form 0 n. 0 n But the denominator of this quotient is 0, which is impossible, since you cannot divide by 0! It is undefined!

THE ZERO POWER Words Numbers The zero power of any number except 0 equals 1. 1000 = 1 (– 7)0 = 1 Algebra a = 1, if a 0

Practice Write the product or quotient as one power 1. n 3 n 4 2. 109 105 t 9 t 7 4. 33 • 32 • 35 3. 5. 8 • 88 = 1. n 7 2. 104 3. t 2 4. 310 5. 89