Mathematics for Technology II (Math 2131)

Durham College, Mathematics
Free
• 36 lessons
• 0 quizzes
• 14 week duration
• Solving Systems of Equations

This unit introduces how to systematically solve a system of equations, namely linear equations. Examples of non-linear systems, including systems of 3 unknowns will be of emphasis.

• Graphs of Trigonometric Functions

The unit focuses primarily on how to graph periodic sinusoidal functions, and how to identify features of a waveform to produce an equation by inspection.

• Polar Coordinate Functions

An introduction to the polar coordinate system.

• Exponents and Radicals

This unit is an extension of what was introduced in Math 1131. To learn how to work with radicals, knowing your exponent laws in crucial. Hence, this unit begins with a thorough review.

• Logarithmic Functions

This chapter introduces you to exponential functions, and how they can be solved using logarithms.

• Trigonometric Identities and Equations
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• Analytic Geometry
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Mathematics for Technology II (Math 2131)

A radical is mathematical way to represent fractional exponents. A radical consists of a radical sign, a quantity under the radical sign called the radicand, and the index of the radical.

If an expression, let’s say a, is raised to the power of ½, that’s the same as taking the square root (√) of a (see below). Finding the square root of a number means (also known as second root) suggests that you’re looking for a number, that when multiplied in itself twice, gives you the radicand.

${a}^{1}{2}}=\sqrt{a}$

What’s the square root (or second root) of the number 4? Can you think of a number, when multiplied in itself twice, gives you 4? If you thought of ±2, you’re right! Given that the answer is an integer, 4 is called a perfect square.

${4}^{1}{2}}=\sqrt{4}=±2$

Similarly, if a is raised to ⅓, that’s the same as taking the cube root (∛) – also known as the 3rd root. Now you’re looking for a number, when multiplied in itself 3 times, gives you the radicand.

${a}^{1}{3}}=\sqrt[3]{a}\phantom{\rule{0ex}{0ex}}{a}^{1}{4}}=\sqrt[4]{a}\phantom{\rule{0ex}{0ex}}⋮\phantom{\rule{0ex}{0ex}}{a}^{1}{n}}=\sqrt[n]{a}$

What’s the cube root (or third root) of the number 27? Can you think of a number, when multiplied in itself 3 times, gives you 27? If you thought of ±3, you’re right! Given that the answer is an integer, 27 is called a perfect cube.

${27}^{1}{3}}=\sqrt[3]{27}=±3$

Notice how an index of 2 isn’t displayed with the root symbol because it is automatically assumed to be a second root. Furthermore, if the numerator of the fractional exponent has a number other than 1, the numerator acts as the exponent to the radicand. This relationship is shown below:

Question:   Evaluate the expression show below:

${8}^{5}{3}}\phantom{\rule{0ex}{0ex}}$

Solution:   First we change the expression to radical form:

${8}^{5}{3}}={\left(\sqrt[3]{8}\right)}^{5}\phantom{\rule{0ex}{0ex}}$

It becomes apparent that you need to evaluate what’s within the parentheses first – the cube root of 8 is 2. Raising 2⁵ is equal to 32.

${2}^{5}=32$

You will need to know how to convert between radical expressions and fractional exponents. The video below shows a few examples:

Another learning expectation is that you know how to simplify algebraic expressions containing fraction exponents without the aid of your calculator. Four examples pertaining to this are shown below: