Power to a power
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When simplifying expressions with exponents raised to exponents, there is a shortcut to avoid expanding it out fully. You can multiply the exponents instead, so (x2)4 becomes x8, as the exponent of 4 means to multiply x2 four times, equivalent to multiplying the exponents 2 and 4 to get 8. Some examples include (x2)3 becoming x6, (t-2)2 becoming t-4, and (m4)1 remaining as m4 since the exponent is 1.
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Power to a Power
(x2)4 can be simplified by taking x2 and multiplying it by itself 4 times, resulting in (x * x * x * x * x * x * x * x) which can be written as x8 by using an exponent to gather the bases. This shortcut of multiplying the powers can be used to simplify other expressions, such as (x2)3 = x6, (t-2)2 = t-4, and (m4)1 = m4.
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This document contains 16 logic and critical thinking problems involving statements, truth tables, logic symbols, validity of arguments, and Euler diagrams. The problems cover topics such as determining logical equivalences, constructing and analyzing truth tables, identifying inverse, converse and contrapositive statements, and determining the validity of arguments using logic rules or diagrams.
C3 test
This document contains 16 logic and critical thinking problems involving statements, truth tables, logic symbols, validity of arguments, and Euler diagrams. The problems cover topics such as determining logical equivalence, constructing and analyzing truth tables, identifying inverse, converse and contrapositive statements, and determining the validity of arguments using logic rules or diagrams.
Intro to Logs
The document introduces logarithms, defining them as the exponent that a fixed number (the base) must be raised to to equal the value. It provides examples of converting between logarithmic and exponential forms, and covers the key properties of logarithms including product, quotient, power, expanding, condensing, and the change of base formula.
Logs
The document introduces logarithms, defining them as the exponent that a base number must be raised to to equal the value. It provides examples of converting between logarithmic and exponential forms, and discusses properties of logarithms such as the product, quotient, and power properties. It also covers expanding, condensing, and changing the base of logarithmic expressions.
Intro to Polynomials
This document discusses polynomial functions and how to graph them. It defines a polynomial as a sum of terms with non-negative integer exponents. Polynomial graphs are smooth curves that may be lines, parabolas, or higher-order curves. To graph a polynomial, one determines the end behavior from the leading term, finds the x-intercepts by setting the polynomial equal to 0, and uses intercepts and test points to plot the graph over intervals. Multiplicity of roots affects whether the graph crosses or is tangent to the x-axis at those points.
Completing the Square
The document explains how to solve quadratic equations by completing the square. It defines a perfect square trinomial as having the form x^2 + bx + c, where c is the square of half of b. It provides steps for completing the square, which involves adding a constant term to both sides of the equation such that the left side becomes a perfect square trinomial that can be factorized. This process results in the solution(s) to the quadratic equation. Two examples demonstrating this process are included.
Graph Quadratics
The document discusses graphing quadratic functions in standard form (y=ax^2 + bx + c). It explains that the graph is a parabola that can open up or down depending on whether a is positive or negative. The line of symmetry for the parabola passes through the vertex and is given by the equation x=-b/2a. The steps to graph are: 1) find the line of symmetry, 2) plug the x-value into the original equation to find the vertex, 3) find two other points and reflect them across the line of symmetry.
Composite functions
Composite functions are formed by taking the output of one function and using it as the input of another function. This is shown notationally as f(g(x)), where the result of g(x) is used as the input for f. Changing the order of the functions changes the result, as the output of the inner function determines the input to the outer function. Examples show evaluating composite functions by substituting the output of the inner function into the outer function and simplifying.
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Power to a power
- 1. When you have a exponent raised to another exponent....like (x2)4 ...there is a shortcut to simplfying the expression. The disabled-expression (x2)4 says take x2 multiply four of them. So what you really have is (x2* x2 * x2 * x2 ). This can be expanded to written as (x * x * x * x * x * x * x * x). When you use an exponent to gather the bases, you have x8 . Or you could have just mutliplied the powers and said (2 * 4) :)<br />Here are some more examples:<br />1. (x2)3 = x6<br />2. (t-2)2 = t-4<br />3. (m4)1 = m4<br />