Learn to evaluate expressions with negative exponents. Exponent Integers Learn to evaluate expressions with negative exponents.

Focus 11 - Learning Goal: The student will be able to work with integer exponents. 4 3 2 1 In addition to 3, student will be able to go above and beyond by applying what they know about working with integer exponents.   The student will be able to work with integer exponents. - Know and apply the properties of exponents. - Simplify numerical expressions with exponents. - Perform operations with scientific notation. With no help the student has a partial understanding of integer exponents. - Is able to use scientific notation to estimate very large or very small numbers. - Interpret scientific notation generated by technology. With help, the student may have a partial understanding of how to work with integer exponents. Even with help, the student is unable to work with integer exponents.

10 2 1 –1 –2 10 • 10 100 = 0.1 = 0.01 ÷10 ÷10 ÷10 ÷10 Look for a pattern in the table to extend what you know about exponents to include negative exponents. Start with what you know about positive and zero exponents.

Evaluate the powers of 10. 1. 10–2 10–2 = 10–2 = = 0.01 2. 10–1 10 = 10 • 10 10–2 = 1 100 = 0.01 2. 10–1 10 = –1 1 10 10 = = 0.1 –1 1 10

Evaluate the powers of 10. 3. 10–8 10–8 = 1 10 • 10 • 10 • 10 • 10 • 10 • 10 • 10 10–8 = = 0.00000001 1 100,000,000 4. 10–9 10–9 = 1 10 • 10 • 10 • 10 • 10 • 10 • 10 • 10 • 10 10–9 = = 0.000000001 1 1,000,000,000

NEGATIVE EXPONENTS Words Numbers Algebra 5–3 = = 125 53 b–n = bn A power with a negative exponent equals 1 divided by that power with it’s opposite exponent. 5–3 = = 1 125 53 b–n = 1 bn The reciprocal of a number is 1 divided by that number. Remember!

Write the reciprocal; change the sign of the exponent. Evaluate. 5–3 Write the reciprocal; change the sign of the exponent. 5 1 3 1 5 • 5 • 5 125 1

Evaluating Products and Quotients of Negative Exponents Evaluate. 1. 2–5 • 23 2 –5+3 Bases are the same, so add the exponents. 2 –2 22 1 Write the reciprocal; change the sign of the exponent. 1 4 Check: 2 • 23 = • 23 = 2 5 1 –5 23 1 4 = 2 • 2 • 2 • 2 • 2 2 • 2 • 2 =

Evaluate. 2. 65 68 Bases are the same, so subtract the exponents. 6 6 Evaluating Products and Quotients of Negative Exponents Continued Evaluate. 65 68 2. Bases are the same, so subtract the exponents. 6 5–8 6 –3 1 63 Write the reciprocal; change the sign of the exponent. 216 1 Check: 6 5 8 6 • 6 • 6 • 6 • 6 • 6 • 6 • 6 6 • 6 • 6 • 6 • 6 = = 216 1

Evaluate. 52 53 3. Bases are the same, so subtract the exponents. 5 2–3 5 –1 1 5 Write the reciprocal; change the sign of the exponent. 1 5 = 53 52 5 • 5 • 5 5 • 5 Check: 5 1 =

Evaluate. Check: 4. 7–6 • 77 Bases are the same, so add the exponents. –6+7 Bases are the same, so add the exponents. 7 1 7 • 7 = = Check: 7 –6 1 6 • 7 7 = 7 • 7 • 7 • 7 • 7 • 7 • 7 7 • 7 • 7 • 7 • 7 • 7 = = 7 7 1