Ninth Grade Mathematics: Surds: Definition, Types & Simplification
Surds represent the exact values of irrational roots that cannot be resolved into terminating or repeating decimals. While approximate decimal calculations are useful in daily life, pure mathematics and scientific engineering demand absolute exactness, which only surd notation can provide.
Definition and Anatomy of a Surd
A surd is defined as an irrational root of a positive rational number. Specifically, if 'a' is a positive rational number and 'n' is a positive integer greater than 1 such that the n-th root of a is not rational, then the expression is called a surd of order n.
Anatomy of a Surd:
Order of the surd (n)
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n _____
/ a <--- Radicand (a rational number)
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Not every radical expression is a surd. For example, sqrt(9) is not a surd because 9 is a perfect square and sqrt(9) = 3 (a rational number). Similarly, cbrt(27) = 3 is not a surd. However, sqrt(2), sqrt(7), cbrt(5), and the 4th-root of 11 are all true surds because their roots are irrational.
Classification and Types of Surds
Surds are classified according to their order, their components, and how they combine with rational numbers:
Surd Type Description Example
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Pure Surd Contains only an irrational root sqrt(18), cbrt(5)
Mixed Surd Product of rational and surd parts 3 * sqrt(2), 5 * cbrt(7)
Like Surds (Similar) Surds with identical order & radicand 4 * sqrt(3) and -7 * sqrt(3)
Compound Surd Algebraic sum of two or more surds sqrt(3) + sqrt(5)
Binomial Surd Compound surd with exactly two terms 2 + sqrt(7), sqrt(5) - sqrt(2)
The order of a surd is indicated by its root index: a square root has order 2 (quadratic surd), a cube root has order 3 (cubic surd), and a fourth root has order 4 (biquadratic surd).
Reducing Surds to Simplest Form
A surd of order n is in its simplest form when the radicand is an integer containing no factor that is a perfect n-th power, and has no fraction under the radical.
Reduction Procedure:
1. Find the largest perfect n-th power that divides the radicand.
2. Rewrite the radicand as a product of this power and the remaining factor.
3. Apply the product rule: n-root(a * b) = n-root(a) * n-root(b).
4. Extract the rational root to the front of the radical.
Example: Simplify sqrt(72)
Largest perfect square factor of 72 is 36:
sqrt(72) = sqrt(36 * 2) = sqrt(36) * sqrt(2) = 6 * sqrt(2) (Mixed surd in simplest form)
Converting a Mixed Surd to an Entire (Pure) Surd
Conversely, any mixed surd can be converted into a pure surd by raising the outer rational coefficient to the power of the surd's order and multiplying it into the radicand:
Convert 3 * cbrt(2) to a pure surd:
Raise the coefficient 3 to the power of the order (3): 3^3 = 27
Place it inside the radicand: cbrt(27 * 2) = cbrt(54)
Comparison of Surds of Different Orders
To compare surds that have different orders (such as comparing sqrt(3) and cbrt(5)), you cannot simply compare the radicands. You must first convert them to equivalent surds of a common order by finding the least common multiple (LCM) of their indices.
Compare sqrt(3) and cbrt(5):
Step 1: Identify orders: order of sqrt(3) is 2; order of cbrt(5) is 3.
LCM of 2 and 3 is 6.
Step 2: Express each surd with order 6 using the property: n-root(a) = (n*k)-root(a^k)
sqrt(3) = 3^(1/2) = 3^(3/6) = 6th-root(3^3) = 6th-root(27)
cbrt(5) = 5^(1/3) = 5^(2/6) = 6th-root(5^2) = 6th-root(25)
Step 3: Compare the radicands:
Since 27 > 25, it follows that 6th-root(27) > 6th-root(25).
Therefore, sqrt(3) > cbrt(5).
Chapter Exercises
Exercise 1: Convert each of the following pure surds into simplest mixed form: sqrt(108), sqrt(245), and cbrt(128).
Exercise 2: Convert the mixed surds 5 * sqrt(3) and 2 * cbrt(4) into entire (pure) surds.
Exercise 3: Arrange the surds sqrt(2), cbrt(3), and 4th-root(5) in ascending order of magnitude by converting them to equivalent surds of a common order.
Solutions and Explanations
Solution 1: For sqrt(108), find the largest perfect square dividing 108, which is 36: sqrt(108) = sqrt(36 * 3) = sqrt(36) * sqrt(3) = 6 * sqrt(3). For sqrt(245), the largest perfect square factor is 49: sqrt(245) = sqrt(49 * 5) = sqrt(49) * sqrt(5) = 7 * sqrt(5). For cbrt(128), we seek the largest perfect cube dividing 128, which is 64: cbrt(128) = cbrt(64 * 2) = cbrt(64) * cbrt(2) = 4 * cbrt(2).
Solution 2: For 5 * sqrt(3), the order of the radical is 2. Square the rational coefficient 5 to get 5^2 = 25, and move it inside the radical: sqrt(25 * 3) = sqrt(75). For 2 * cbrt(4), the order of the radical is 3. Cube the coefficient 2 to get 2^3 = 8, and move it inside the radical: cbrt(8 * 4) = cbrt(32).
Solution 3: The orders of the three surds are 2, 3, and 4. The least common multiple of 2, 3, and 4 is 12. Convert each surd to order 12: for sqrt(2) = 2^(1/2), multiply index and exponent by 6: 12th-root(2^6) = 12th-root(64). For cbrt(3) = 3^(1/3), multiply index and exponent by 4: 12th-root(3^4) = 12th-root(81). For 4th-root(5) = 5^(1/4), multiply index and exponent by 3: 12th-root(5^3) = 12th-root(125). Comparing the radicands gives 64 < 81 < 125, which corresponds to 12th-root(64) < 12th-root(81) < 12th-root(125). Therefore, the surds in ascending order are sqrt(2) < cbrt(3) < 4th-root(5).