In the past, you have worked with square roots, cube roots, and the process of estimating roots. Let's refresh our memories, and take a closer look.
In Algebra, we will also be working primarily with square roots and cube roots.
But instead of estimating roots, we will be simplifying roots.
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Radicals:

 

A radical expression is one which contains a root (square root, cube root, etc.).
        radparts    rad rules

In Algebra 1, radical expressions are primarily limited to square root (a = 2) expressions and cube root (a = 3) expressions.

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Square Roots:

 


On this page, the radicand will be non-negative. No negatives under the radical.]
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.

 
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To take the square root of a number,
is to un-do (or reverse) the squaring process.

Finding the square root of a number is the inverse operation of squaring the number.
4 squared = 16
square root of 16 = 4
 sqsqrt
Squaring: 92 = 81      Square rooting: squareroot81

star   squareroot+-
It is important to remember that if there is NO SIGN in front of a square root symbol, you are dealing with the positive answer only. The positive sign is implied to be there, indicating the principal square root.
If a negative symbol is in front of a square root symbol, the answer will be negative.

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The square root of a value is a quantity which, when squared, equals the radicand (the number under the square root symbol.) For example, the square root of 16 could be either + 4 or -4, since both, when squared, equal 16.

It is understood, however, that the square root (radical) symbol denotes only the positive root, called the "principal square root". yel 16

When solving the equation x2 = 25, you are searching for both solutions: +5 and -5. So, we write: squared    


radsigns

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Cube Roots:

 


"Cubing" is the same as raising to the power of 3,
denoted by an exponent of 3.

To take the cube root of a number,
is to un-do (or reverse) the cubing process.
Finding the cube root of a number is the inverse operation of cubing the number.
2 cubed = 8
cube root of 8 = 2
 sqsqrt
Cubing: 43 = 64      Cube rooting: cuberoot64
The radical symbol used for a cube root requires
an index (root) of 3, which must always appear in the symbol.

star  cubes2
When working with cube roots, the sign inside the cube root symbol
will determine the sign of the answer.

You may see a negative number under a cube root symbol.
RAD543
RAD643A
Perfect Squares
0 = 0 x 0
1 = 1 x 1
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
r1
r2
r3
r4
r5
r6
r7
r8
r9
r10
r12
r13
r14
r15

Perfect Cubes
0 = 0 x 0 x 0
1 = 1 x 1 x 1
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
243 = 7 x 7 x 7
512 = 8 x 8 x 8
729 = 9 x 9 x 9

Cube Roots
cuberoot0
cuberoot1
21
22
23
24
morecuberoot

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For calculator help with radicals
click here.
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Remembering Estimating Radicals:

 



If you are dealing with a perfect square (or perfect cube) under a radical (index 2 or 3), the numerical equivalent is usually obvious.
r7       22
But if the number under the radical (the radicand) is not a perfect square, the numerical value of the radical will be an irrational number, yielding a non-ending decimal estimate.

You have looked at such estimates in two forms:
1. Estimate the value using a calculator: The calculator will give a non-ending decimal value as an approximation of the radical's value.

2. Estimate the value by sandwiching: The radical is placed between two consecutive perfect square roots to achieve an approximation of the value of the radical.

sq1623
sq423
This estimate can be further investigated (by hand) using a "Guess and Check" method.
See more about "Estimating Square Roots" under Junior Math.

 
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