Bell Work: Solve. Check your work. 4x + 10 + x = 100

Answer: x = 18

Lesson 51: Negative exponents, scientific notation for small numbers

Recall from the laws of exponents, that we subtract the exponents when dividing exponential expressions that have the same base. x ÷ x = x 5 3 2

We can understand this law by applying what we know about exponents and division. x = x  x  x  x  x = x x x  x  x 5 2 3

Now consider the result when we reverse the numbers in the division Now consider the result when we reverse the numbers in the division. Following the laws of exponents, the exponent of the quotient is negative. x ÷ x = x (because 3 – 5 = -2) 3 5 -2

We will apply what we know about exponents and division to understand the meaning of x . x = x  x  x = 1 x x  x  x  x  x x -2 3 5 2

By performing the division we find that x means 1/x By performing the division we find that x means 1/x . We see that x is the reciprocal of x . This fact is another law of exponents. -2 2 -2 2

Law of Exponents for Negative Exponents: x = 1/x

Applying this law to powers of 10, we see the following pattern Applying this law to powers of 10, we see the following pattern. Note that 10 = 1. 10 = 100 10 = 10 10 = 1/10 or .1 10 = 1/100 or 0.01 2 1 -1 -2

Example: Use the law of exponents to find the product of 10 and 10 . 2 -2

Answer: 1

Very small numbers may exceed the display capabilities of a calculator Very small numbers may exceed the display capabilities of a calculator. One millionth of one millionth is more than zero, but it is a very small number. On a calculator it may be read as 1 x 10 . -12

Which of the following does not equal 10 ? 1/10 1/1000 0.001 -1000 Practice: Which of the following does not equal 10 ? 1/10 1/1000 0.001 -1000 -3 3

Answer: d) -1000

Practice: Solve 10  10 = 10 -2 -4

Answer: -6

Practice: Solve 10 /10 = 10 -2 -4

Answer: 2

Practice: Simplify 2 -3

Answer: 1/8

Practice: Simplify (-2) 3

Answer: -8

Practice: Simplify 3  3 2 -2

Answer: 1

Practice: Express with positive exponents and simplify: 2x yx y z -1 2 -2

Answer: 2xz y

We use negative powers of 10 to write small numbers in scientific notation. By small numbers we mean numbers between 0 and 1.

Example: Write this number in standard form. 1.5 x 10 -3

Answer: 1.5 x 0.001 = 0.0015

Practice: The diameter of a red blood cell is about 0. 000007 meters Practice: The diameter of a red blood cell is about 0.000007 meters. Write that number in scientific notation.

Answer: 7 x 10 -6

Practice: Compare: 1 x 10 -10 -6

Answer: >

HW: Lesson 51 #1-30