 Examining Radicals MathBitsNotebook.com Terms of Use   Contact Person: Donna Roberts A radical expression contains a root (square root, cube root, etc.).  At the Junior level, radical expressions will be involved in working with the Pythagorean Theorem and with Volume, and will be limited to square root (a = 2) expressions and cube root (a = 3) expressions. The value of R (the radicand) will be limited to positive values.  Square of a Number
Finding the Square of a Number:
• Squaring a number is multiplying a number times itself.
 square of 7 = 7 x 7 = 72 = 49 square of n = n x n = n2 • Perfect squares are the squares of integers. (Examples shown at the right.) You may also find definitions that state a perfect square is the square of any rational number. Squaring a number creates a power (exponent) of 2 (or an even power).

When numbers are "squared", even powers (exponents) are created.  Notice: square of 3 = +9 square of -3 = +9 Same result !  Square Roots:
Square roots have an index value of two. When you see a radical with no index listed, it is assumed to be an index of two, a square root. Finding the square root of a number is the inverse operation of squaring the number. 4 squared = 16 square root of 16 = 4 Finding the Square Root of a Number:
• Square root of a number, n, is the number that gives n when multiplied times itself. Now, you might ask, "can't the answer also be -5, since -5 x -5 = 25?"
• When no sign is indicated in front of , it is referred to as the
• If the negative answer is expected, you will see .
• If both positive and negative answers are expected, you will see .
For example, when solving the equation x2 = 25, you are searching for both solutions: +5 and -5. So, we write: • Examples of square roots of perfect squares
are shown at the right.

The value is
not a real number and is not studied at this level.
Note:   Cube of a Number
Finding the Cube of a Number:
• Cubing a number is multiplying a number times itself three times.
 cube of 7 = 7 x 7 x 7 = 73 = 343 cube of n = n x n x n = n3 • Perfect cubes are the cubes of integers. (Examples shown at the right.) Cubing a number creates a power (exponent) of 3 (or multiple of 3).

When numbers are "cubed", powers that are multiples of 3 are created. Notice: cube of 3 = +27 cube of -3 = -27 NOT the same !   Cube Roots:
Cube roots have an index value of three. Finding the cube root of a number is the inverse operation of cubing the number. 2 cubed = 8 cube root of 8 = 2 Finding the Cube Root of a Number:
• Cube root of a number, n, is the number that gives n when multiplied times itself three times. When square rooting, we saw that the result may be both positive and negative. That is NOT the case when cube rooting. -2 x -2 x -2 = -8 ≠ +8

• Examples of cube roots of perfect cubes
are shown at the right. When working with cube roots, it is easy to forget to write the index value of 3 on the symbol. Be careful! Without the 3 written in the cube root symbol, your answer will be incorrect, as it represents a different value. The cube root of 5 does not equal the square root of 5.

 Perfect Squares 4 = 2 x 2 9 = 3 x 3 16 = 4 x 4 25 = 5 x 5 36 = 6 x 6 49 = 7 x 7 64 = 8 x 8 81 = 9 x 9 100 = 10 x 10 121 = 11 x 11 144 = 12 x 12 169 = 13 x 13 196 = 14 x 14 225 = 15 x 15 Square Roots              Perfect Cubes 8 = 2 x 2 x 2 27 = 3 x 3 x 3 64 = 4 x 4 x 4 125 = 5 x 5 x 5 216 = 6 x 6 x 6 343 = 7 x 7 x 7 512 = 8 x 8 x 8 729 = 9 x 9 x 9 Cube Roots      For calculator help with radicals click here.  Radicals on the Number Line
Three situations for graphing radical values on a number line:
1) The square root of a perfect square integer will be an integer whose
exact value is easy to plot. (An integer which is a rational number.)
2) The cube root of a perfect cube integer will be an integer whose
exact value is easy to plot. (An integer which is a rational number.)
3) If you are not dealing with option 1 or 2, you will need to estimate the value of the square root or cube root before plotting. These plots will be "
estimates" or "approximations" of the actual values. (Irrational numbers.) The rational numbers are plotted as exact values.
The irrational values are plotted as approximations.   Estimating Square Roots without a Calculator
To estimate the value of the square root of a number, we are going to squeeze the square root between the square roots of two perfect squares.

Example: Estimate to the nearest tenth.
Remember that we are looking for a number, when squared, that will be approximately 12.
 1. Between which two perfect squares does 12 fall? 1. 12 falls between 9 and 16         9 < 12 < 16 2. Establish as square roots. 2. 3. falls between 3 and 4. 3. 4. Let's guess at 3.5 and see where we stand. 4.   3.52 = 12.25 (bit too big) 5. Let's try 3.4. 5.   3.42 = 11.56 (bit too small) 6. Let's check the hundredths place digit by trying 3.45. This will tell us whether the rounded answer is 3.4 or 3.5. 6.   3.452 = 11.9025 (still a bit too small) 7. When rounded to the nearest tenth, we now know our answer will round to 3.5. 7.  