Question

What are like surds and unlike surds? Identify and write the set of like surds in the following groups:
A. \[\left\{ {\sqrt 8 ,\sqrt {12} ,\sqrt {20} ,\sqrt {54} } \right\}\]
B. \[\left\{ {\sqrt {50} ,\sqrt[3] {{54}},\sqrt[4] {{32}}} \right\}\]
C. \[\left\{ {\sqrt 8 ,\sqrt {18} ,\sqrt {32} ,\sqrt {50} } \right\}.\]

Answer 1

Hint: This question involves the arithmetic operations like addition/ subtraction/ multiplication/ division. We need to know the definition of like surds and unlike surds. Also, we need to know how to find radicand and index in the given term. Also, we need to know the square and square root values for basic terms.

Complete step by step solution:
In this question, we would find which is like surds and which is unlike surds in the given sets.
Before that, we would find the definition of like surds and unlike surds.
Like surds are those surds that have the same radicand (In \[\sqrt a \] \[ - \] \[a\] is the radicand).
Unlike surds are those surds which have different radicand.
Let’s solve the given problem,
A. \[\left\{ {\sqrt 8 ,\sqrt {12} ,\sqrt {20} ,\sqrt {54} } \right\}\]
Here, \[\sqrt 8 \] can be written as \[\sqrt {2 \times 2 \times 2} = 2\sqrt 2 \]
 \[\sqrt {12} \] Can be written as \[\sqrt {2 \times 2 \times 3} = 2\sqrt 3 \]
 \[\sqrt {20} \] Can be written as \[\sqrt {2 \times 2 \times 5} = 2\sqrt 5 \]
 \[\sqrt {54} \] Can be written as \[\sqrt {3 \times 3 \times 6} = 3\sqrt 6 \]
So, we get \[\left\{ {\sqrt 8 ,\sqrt {12} ,\sqrt {20} ,\sqrt {54} } \right\}\] \[ = \left\{ {2\sqrt 2 ,2\sqrt 3 ,2\sqrt 5 ,3\sqrt 6 } \right\}\]
Here we have different radicands, so the set \[\left\{ {\sqrt 8 ,\sqrt {12} ,\sqrt {20} ,\sqrt {54} } \right\}\] is unlike surds.

B. \[\left\{ {\sqrt {50} ,\sqrt[3] {{54}},\sqrt[4] {{32}}} \right\}\]
Here, \[\sqrt {50} \] can be written as \[\sqrt {5 \times 5 \times 2} = 5\sqrt 2 \]
 \[\sqrt[3] {{54}}\] Can be written as \[\sqrt[3] {{3 \times 3 \times 3 \times 2}} = 3\sqrt[3] {2}\]
 \[\sqrt[4] {{32}}\] Can be written as \[\sqrt[4] {{2 \times 2 \times 2 \times 2 \times 2}} = 2\sqrt[4] {2}\]
So, we get \[\left\{ {\sqrt {50} ,\sqrt[3] {{54}},\sqrt[4] {{32}}} \right\}\] \[ = \left\{ {5\sqrt 2 ,3\sqrt[3] {2},2\sqrt[4] {2}} \right\}\]
Here we have the same radicands, so the set \[\left\{ {\sqrt {50} ,\sqrt[3] {{54}},\sqrt[4] {{32}}} \right\}\] is like surds.

C. \[\left\{ {\sqrt 8 ,\sqrt {18} ,\sqrt {32} ,\sqrt {50} } \right\}.\]
Here, \[\sqrt 8 \] can be written as \[\sqrt {2 \times 2 \times 2} = 2\sqrt 2 \]
 \[\sqrt {18} \] Can be written as \[\sqrt {3 \times 3 \times 2} = 3\sqrt 2 \]
 \[\sqrt {32} \] Can be written as \[\sqrt {2 \times 2 \times 2 \times 2 \times 2} = 4\sqrt 2 \]
 \[\sqrt {50} \] Can be written as \[\sqrt {5 \times 5 \times 2} = 5\sqrt 2 \]
So, we get \[\left\{ {\sqrt 8 ,\sqrt {18} ,\sqrt {32} ,\sqrt {50} } \right\}.\] \[ = \left\{ {2\sqrt 2 ,3\sqrt 2 ,4\sqrt 2 ,5\sqrt 2 } \right\}\]
Here we have the same radicands, so the set \[\left\{ {\sqrt 8 ,\sqrt {18} ,\sqrt {32} ,\sqrt {50} } \right\}.\] is like surds.
So, the final answer is,
A. \[\left\{ {\sqrt 8 ,\sqrt {12} ,\sqrt {20} ,\sqrt {54} } \right\}\] Is unlike surds.
B. \[\left\{ {\sqrt {50} ,\sqrt[3] {{54}},\sqrt[4] {{32}}} \right\}\] Is like surds.
C. \[\left\{ {\sqrt 8 ,\sqrt {18} ,\sqrt {32} ,\sqrt {50} } \right\}.\] Is like surds.

Note: Note that if the surds have the same radicand which is called like surds and if the surds have different radicand which is called, unlike surds. Also, note that the square and square root can be cancelled each other and the cube and cubic root can be cancelled each other. Note that the term inside the radical sign is called the radicand.