Introduction
In this lab, you will review the basic algebra skills necessary for this course. Be sure to get help early if you have difficulty with these problems!
Solving Algebraic Equations
An equation is a mathematical statement in which one expression is equal to another.
Example:
4 + 7 = 11
Algebraic equations involve an unknown variable.
Example:
5x + 8 = 18
Solving an equation means finding the possible value (or values) of the unknown variable that would satisfy the original equation. In solving an equation, it is most important to keep in mind that any operation performed on one side must also be performed on the other side.
Example:
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Example:
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Rules for Exponents
There are three primary rules to keep in mind when working with exponents:
Rule
Example 1
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Example 2
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Note: With the exception of scientific notation, all final answers MUST be expressed in terms of POSITIVE exponents. Equation Manipulation
Equ tions descri ing the l ws of physics re often expressed in “st nd rd” form. It is often necess ry, however, to manipulate these equations in order to solve for (or isolate) a particular variable.
See Appendix III for additional examples and practice problems.
Example: Solve for A:
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Example: Solve for Ti:
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Example: Solve for a:
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Scientific Notation
Very large and very small numbers are often expressed in scientific notation. Numbers in this notation are expressed as the product of a number (whose absolute value is between one and ten) and a power of ten. In m n other words, numbers in scientific notation fit the form n
, where and m is an integer.
See Appendix II for additional examples, practice problems, and information on this topic – including how to properly enter numbers in scientific notation on your calculator.
Examples:
3
3,800,000,000 = 3.8 109
0.000 026 = 2.6 105
Moreover, scientific notation helps to clarify precision and significant figures. This topic will be thoroughly explained in Lab 1.
Large numbers like 43 million contain many digits mostly trailing zeros. Small numbers like 52 billionths also contain many digits mostly leading zeroes. Scientific notation allows us to write down a large or small number in terms of powers of ten without all of the zeros.
If a number’s solute v lue is l rger th n , count the number of place values you must move the decimal to the left to make it a number between 1 and 10, and use that number as the power of ten.
Examples:
34,000,000 = 3.4 × 107
545,400,000,000,000 = 5.454 × 1014
4,720,000 = 4.72 × 106
Note: In the last example above, if the zero in the thousands place were significant (i.e. that place value was measured and the digit just happened to be zero), the number MUST be written in scientific notation in order to avoid ambiguity regarding which digits are significant (i.e. directly or indirectly measured). The meaning of
“signific nt figures” will e fully ddressed in Lab 1.
Example:
4,720,000 = 4.720 × 106
If a number’s solute v lue is smaller than 1, count the number of place values you must move the decimal to the right to make it a number between 1 and 10, and use that number as the negative power of ten.
Examples:
0.000 065 4