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Mathematics for Technology II (Math 2131)

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Study Force Academy
Durham College, Mathematics
Free
  • 36 lessons
  • 0 quizzes
  • 14 week duration

Introduction to Radicals

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.

Anatomy of a 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.

a12=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.

412=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.

a13=a3a14=a4a1n=an

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.

2713=273=±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:

amn=amn   or   anma23=a23   or   a32

Question:   Evaluate the expression show below:

853

Solution:   First we change the expression to radical form:

853=835

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.

25=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:

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