Radical expressions in algebra involve roots and exponents. They have three components: index, root, and radical. Simplifying involves dividing into factors, adding requires the same index and radical, and multiplying uses the distributive property. Equations are solved by eliminating radicals through squaring.
A radical expression in algebra is an expression that includes a radical, or root. These are the inverse operations of exponents, or powers. Radical expressions include added roots, multiplied roots, and expressions with variables and constants. These expressions have three components: the index, the root and the radical. The index is the degree taken, the radical is the derived root, and the radical is the symbol itself.
By default, a radical sign symbolizes a square root, but by including different indices on the radical, you can take cube roots, fourth roots, or any root of whole numbers. Radical expressions can include numbers or variables below the radical, but the fundamental rules remain the same regardless. To work with radicals, expressions must be in their simplest form; this is achieved by removing the factors from the rooting.
The first step in simplifying radicals is to divide the radical into the factors needed to equal the number. Thus, all perfect square factors must be placed to the left of the radical. For example, √45 can be expressed as √9*5 or 3√5.
To add radical expressions, the index and the radical must be the same. After these two requirements have been met, numbers outside the radical can be added or subtracted. If the radicals cannot be simplified, the expression must remain in a different form. For example, 2+√5 cannot be simplified because there are no factors to separate. Both terms are in their simplest form.
Multiplying and dividing radical expressions work using the same rules. Products and quotients of radical expressions with similar indices and radicalands can be expressed under a single radical. The distributive property works the same way as for integer expressions: a(b+c)=ab+ac. The number outside the brackets must be multiplied in turn by each term inside the brackets, maintaining the operations of addition and subtraction. After all terms within the distributive brackets have been multiplied, the radicals need to be simplified as usual.
Radical expressions that are part of an equation are solved by eliminating the radicals according to the index. Normal radicals are eliminated by squaring; therefore, both sides of the equation are square. For example, the equation √x=15 is solved by squaring the square root of x on one side of the equation and 15 on the right, yielding a result of 225.
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