• Section 5.2 in the textbook

– Expressions with negative exponents

– Simplifying more complex exponential

expressions

–Writing numbers in scientific notation

– Scientific notation to standard form

Exponents

• Consider x^{2} / x^{6}

x^{-4} by the quotient rule

x·x / x·x·x·x·x·x

1 / x^{4}

• Usually, we do NOT leave an expression

with a negative exponent

• Flipping an exponent AND its base from

the numerator into the denominator (or

vice versa) reverses the sign of the

exponent

– The sign of the exponent DOES NOT affect

the sign of the base!

–Whenever using the quotient rule, the initial

result goes into the numerator

Exponents (Example)

**Ex 1:** Simplify – leave NO negative exponents:

Exponential Expressions

• Product Rule:

• Power Rule:

– Power of a Product:

– Power of a Quotient:

• Quotient Rule:

Exponential Expressions

• When solving more complicated

expressions:

– Simplify inside of the parentheses using the

product and quotient rules if possible

– Apply the power (if one exists)

• Use the power rule with the exponents

• Evaluate the bases as with normal numbers

– If necessary, write the final answer with

positive exponents

Exponential Expressions (Example)

**Ex 2:** Simplify – leave NO negative

exponents:

**Ex 3:** Simplify – leave NO negative

exponents:

**Ex 4:** Simplify – leave NO negative

exponents:

Notation

**• Scientific Notation: **any number in the

form of a x 10^{b} where -10 < a < 10, a ≠ 0

and b is an integer

– Used to write extreme numbers (large or

small) in a compact format

• To write a number in scientific notation:

– One non-zero number to the left of the

decimal point – the rest to the right

– Determine where to place the decimal point:

• Count how many places the decimal point is

moved

• If the original number (without the sign) is greater

than 1, b (the exponent) is positive

• If the original number (without the sign) is less than

1, b is negative

Notation (Example)

**Ex 5:** Write each in scientific notation:

a) -238.41

b) 0.00584

c) 0.018

Form

**• Standard Notation:** writing a number in

scientific notation without the power of ten

– Take the decimal and move it:

• To the right if b (the exponent) is positive

• To the left if b (the exponent) is negative

• Fill in empty spots with zeros

Form (Example)

**Ex 6: **Write in standard notation:

Notation

• Multiply or divide the numbers as normal

• Use the Product or Quotient Rules to

simplify the power of tens

• Write the final answer in scientific notation

Scientific Notation (Example)

**Ex 7:** Simplify and write as a single number

in scientific notation:

• After studying these slides, you should know

how to do the following:

– Simplify expressions containing negative exponents

– Apply the exponent rules to simplify more complex

exponential expressions

– Writing numbers in scientific notation

– Converting scientific notation to standard form

– Multiply or divide using scientific notation

• Additional Practice

– See the list of suggested problems for 5.2

• Next lesson

– Introduction to Polynomials (Section 5.3)