Radicals, also known as surds, are mathematical expressions that represent the roots of numbers. These expressions include a square root symbol (√) and contain either a number or a variable under the radical. The radical symbol represents the mathematical process of finding the root of a number.
The most frequently used radical is the square root, which is represented by the √ symbol. To find the square root of a number, we look for a value that, when multiplied by itself, gives the original number. For instance, because 3 multiplied by itself equals 9, the square root of 9 is 3.
In general, for any positive number a, the square root is written as √a. It's important to remember that every positive number has two square roots: one positive (√a) and one negative (−√a), because (−√a) × (−√a) = a as well.
Radicals can also represent other types of roots, such as cube roots (∛) or higher-order roots. The cube root of a number b, symbolized by ∛b, is the value that, when multiplied by itself three times, equals b.
Whenever possible, radicals should be simplified. For example, √9 simplifies to 3, and ∛27 simplifies to 3.
Operations involving radicals include addition, subtraction, multiplication, and division. When working with these operations, it is essential to follow the appropriate rules to ensure accurate results.
Examples:
Square root of a: \( \sqrt{a} \) or \( a^{\frac{1}{2}} \)
Cube root of b: \( \sqrt[3]{b} \) or \( b^{\frac{1}{3}} \)
Finding the nth root of a number
If \( a = (x \times x \times x \times \ldots {n times}) \cdot (y \times y \times y \times \ldots {n times}) \), then:
\( \sqrt[n]{a} = \sqrt[n]{(x \times x \times \ldots)^n \cdot (y \times y \times \ldots)^n} \)
Example 1:
\(
\sqrt[3]{64} = \sqrt[3]{(2 \times 2 \times 2)(2 \times 2 \times 2)} = \sqrt[3]{2^3 \times 2^3} = \sqrt[3]{4^3} = 4
\)
Example 2:
\(
\sqrt[3]{1728} = \sqrt[3]{(2 \times 2 \times 2)(2 \times 2 \times 2)(3 \times 3 \times 3)} = \sqrt[3]{2^3 \times 2^3 \times 3^3} = 2 \times 2 \times 3 = 12
\)
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