Radicals, also known as surds, are mathematical expressions that represent roots of numbers. These expressions involve a radical symbol (√) and a number or a variable under the radical. The radical symbol signifies the operation of finding the root of a number.
The most common radical is the square root, denoted by √. When we take the square root of a number, we are looking for a value that, when multiplied by itself, gives us the original number. For instance, the square root of 9 is 3, because 3 × 3 = 9.
In general, if we have a positive number 'a', the square root of 'a' is denoted as √a. It's important to note that every positive number has two square roots: a positive square root (√a) and a negative square root (-√a), because (-√a) × (-√a) = a as well.
Radicals can also represent other roots, such as cube roots (∛) or higher-order roots. The cube root of a number 'b' is denoted by ∛b and is the value that, when multiplied by itself three times, gives us the original number 'b'.
When working with radicals, it's helpful to simplify them if possible. For instance, √9 can be simplified to 3, and ∛27 can be simplified to 3 as well.
Operations involving radicals include addition, subtraction, multiplication, and division. When combining or manipulating expressions with radicals, it's important to follow specific rules to ensure accurate calculations.
Ex:- Square root of a is \(\sqrt{a}\) or \(a^{\frac{1}{2}}\)
Cube root of b is \(\sqrt[3]{b}\) or \(b^{\frac{1}{2}}\)
Find the nth root of a number
If \(a=(x\times x\times x\times .......n \ times)(y\times y\times y\times .......n \ times)\)
\(\sqrt[n]{a}=\sqrt[n]{(x\times x\times x\times .......n\ times)(y\times y\times y\times .......n \ times)}\)
\(\sqrt[3]{64}=\sqrt[3]{(2\times2\times2)(2\times2\times2)}\)
\(=\sqrt[3]{2^{3}\times 2^{3}}\)
\(=\sqrt[3]{4^{3}}=4\)
\(\sqrt[3]{1728}=\sqrt[3]{(2\times2\times2)(2\times2\times2)}\)
\(=\sqrt[3]{2^{3}\times 2^{3} \times 3^{3}}\)
\(=2 \times 2 \times3=12\)
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