### Operations and scientific notation

Before we begin, let’s restate the properties of exponents so that we have them available to use in the examples.

Properties of Exponents

• Zero Exponent Property                            $$\Large {a^0} = 1, a \ne 0$$
• Negative Exponent Property                    $$\Large {a^{ - b}} = \frac{1}{{{a^b}}}, a \ne 0$$
• Product of Powers Property                     $$\Large {a^b} \cdot {a^c} = {a^{b + c}}, a \ne 0$$
• Quotient of Powers Property                   $$\Large \frac{{{a^b}}}{{{a^c}}} = {a^{b - c}}$$     $$a \ne 0$$
• Power of a Product Property                    $$\Large {a^c} \cdot {b^c} = {\left( {ab} \right)^c}, a,b \ne 0$$
• Power of a Quotient Property                  $$\Large \frac{{{a^c}}}{{{b^c}}} = {\left( {\frac{a}{b}} \right)^c}, a,b \ne 0$$
• Power of a Power Property                       $$\Large {\left( {{a^b}} \right)^c} = {a^{bc}}$$
• Rational Exponent Property                      $$\Large {a^{\frac{1}{b}}} = \sqrt[b]{a}, b \ne 0$$
•                                                                      $$\Large {a^{\frac{c}{b}}} = \sqrt[b]{{{a^c}}} = {\left( {\sqrt[b]{a}} \right)^c}$$

Now we want to use these properties to solve problems involving scientific notation.  There is nothing to be worried about here.  Just remember that in scientific notation, the $$\times$$ stands for multiplication, just like it did back in third and fourth grade.  Don’t let the look of the problem intimidate you.  Let’s jump straight to an example.

Example:  Simplify

$$\large \frac{{4 \times {{10}^{ - 2}}}}{{3.01 \times {{10}^{ - 2}}}}$$

Solution:  We use the properties of exponents.

 $$\large \frac{{4 \times {{10}^{ - 2}}}}{{3.01 \times {{10}^{ - 2}}}} = \frac{4}{{3.01}} \cdot \frac{{{{10}^{ - 2}}}}{{{{10}^{ - 2}}}}$$ $$\large = \frac{4}{{3.01}} \cdot {10^{ - 2 - \left( { - 2} \right)}}$$ Quotient of Powers Property $$\large = \frac{4}{{3.01}} \cdot {10^0}$$ $$\large = \frac{4}{{3.01}} \cdot 1,$$ Zero Exponent Property $$= 1.329$$ Rounded to three decimal places $$= 1.329 \times {10^0}$$ Scientific notation required,and $${10^0} = 1$$

Now we did a few extra steps in the problem above, but it is important to take our time in the beginning, and make sure we are doing the problems the right way.

Example:  Simplify

$$\left( {8.8 \times {{10}^4}} \right)\left( {8 \times {{10}^{ - 6}}} \right)$$

Solution:  We use the properties of exponents.

 $$\left( {8.8 \times {{10}^4}} \right)\left( {8 \times {{10}^{ - 6}}} \right) = \left( {8.8} \right)\left( 8 \right)\left( {{{10}^4}} \right)\left( {{{10}^{ - 6}}} \right)$$ $$= \left( {8.8} \right)\left( 8 \right)\left( {{{10}^{4 + \left( { - 6} \right)}}} \right),$$ Product of Powers Property $$= \left( {8.8} \right)\left( 8 \right)\left( {{{10}^{ - 2}}} \right)$$ $$= \left( {8.8} \right)\left( 8 \right)\left( {{{10}^{ - 2}}} \right)$$ $$= 7.04\left( {10} \right)\left( {{{10}^{ - 2}}} \right),$$ Working toward scientific notation $$= 7.04\left( {{{10}^{1 + \left( { - 2} \right)}}} \right),$$ Product of Powers Property $$= 7.04 \times {10^{ - 1}},$$ Scientific Notation 4682 x

Simplify. Write each answer in scientific notation.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question: Watch bellow how to solve this example: 3620 x

Simplify. Write each answer in scientific notation.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question: Watch bellow how to solve this example: 2830 x

Simplify. Write each answer in scientific notation.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question: Watch bellow how to solve this example:

### Geometry

Circles
Congruent Triangles
Constructions
Parallel Lines and the Coordinate Plane
Properties of Triangles

### Algebra and Pre-Algebra

Beginning Algebra
Beginning Trigonometry
Equations
Exponents
Factoring
Linear Equations and Inequalities
Percents
Polynomials