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Mathematics is a broad discipline involving a lot of concepts and topics, and one of the main challenges students face while pursuing this discipline is to present their ideas. The constant struggle of juggling between understanding the equation and determining results is somewhat tricky. The explains the coexistence of the surprising mix of struggle and competence. One such concept in math is a radical form that involves several components to figure out the results. While students stand confused whenever they encounter radical form problems, it is often that radical and fractional exponents are the alternative ways of saying things. If you are also in the same dilemma and confuse yourself with the concepts, read on to gain clarity.

A radical is the opposite of an exponent in mathematics and is represented with the symbol '√', also known as the root. This can either be a square root or a cube root. And the number before the symbol or the radical is known to be the index number or degree. A radical form is an expression that contains a radical sign '()' and is said to be in the reduced form of the radicand, which is the number under the radical sign and does not contain any perfect squares or cubes. This number is generally a whole number and is represented as an exponent that cancels out the radical.

The symbol '√' expresses a root of a number and is known as the radical and is read as x radical or the n^{th }root of x. The horizontal line that covers the number is called the vinculum, and the number under the horizontal line is called the radicand. The number represented as n written before the radical is called the index or degree.

The simplest radical form means to simplify a radical so that there are no more square, cube, 4^{th} or other roots left to find. Additionally, it also means removing any radicals in the fraction's denominator. However, before you learn to simplify the radicals, you need to know specific rules about them. These rules are primarily dependent on integral exponents and fractional exponents.

Exponents provide a very convenient way of writing large and small numbers. They also come in very handy when trying to make algebra easier since the equations are more compact and complex. Here is a general definition of an integral exponent.

a^{m} means "multiply m lots of a together."

*a*^{m }*= a *x* a *x* a *x* a …. *x* a* (m means a lots of a)

Example:

Y^{5} = y × y × y × y × y

In this equation, there are 5 lots of y being multiplied together.

2^{4} = 2 × 2 × 2 × 2 = 16

There are 4 lots of 2 multiplied together.

These can be used instead of the radical sign (√). Fractional exponents are used because they are way more convenient, and they can make algebraic equations easier to solve. The n-th root of a number can be written using the power 1/n as:

A^{1/n} = n√a

This means that the n-th root of a, when multiplied by itself n number of times, gives us the answer of a.

Example:

The cube root of 8 is 2 (2^{3} = 8)

We can state the cube root of 8 as:

8^{1/3}

Or,

3√8

Hence the following three numbers are equivalent:

8^{1/3 }= 3√8 = 2

The laws of radicals are directly derived from the laws of exponents by using the equation 'a m n = a m n'. These laws simplify simplification and are essential to reducing the radical to its simplest form. The laws of radical are obtained directly from the laws of exponents utilizing:

n√a^{m} = a^{m/n}

The laws of radicals can be stated as follows:

If n is even, we can assume that a, b ≥ 0.

The laws are as follows:

- Simplification Of Radicals

To find out the values, it is essential to simplify the radical and use the following operations:

- First, remove the perfect n-th power from the radicand using n√ab = n√a n√b.

- Reduce the index of the radical

- Rationalization of denominators

Here fractions are removed from the radical sign by rationalizing the denominator. For example, to rationalize the denominator of the radical order of n, you can multiply the numerator and the denominator of the radicand a quantity that will make the denominator a perfect n-th power and remove the denominator under the radical sign.

Examples:

- Addition and Subtraction of Radicals

Before adding or subtracting radicals, it is vital to reduce them to the simplest form. After which, radicals can be added or subtracted similarly to other terms.

Example:

- Multiplication of Radicals

To multiply two or more radicals that have the same index value, you can use n√a n√b = n√ab

Example:

Follow this example to multiple radicals with different indices and fractional exponents:

- Division of Radicals

Use the following formula to divide two radicals having the same index n√a/n√b =n√a/b

Example:

To divide radicals with different fractional exponents and indices, follow this example:

The square root of 12 in the radical form is represented as √12, equal to 2√3. Since this cannot be simplified further, the roots are known as surds. Besides describing the value of root 12 in the radical form, it can also be written in the decimal form as √12 = 3.464. However, the question still arises about how to simplify root 12 and find out the value.

12 is not a perfect square number like 2, 3, 5, 6, 24 etc. Hence the simplification can be a little typical. However, for numbers like 4, 16, 25, 81 etc., it is easier to estimate square roots since they are perfect squares. The Square of a number can be achieved by multiplying the number by itself. Thus, to answer how to find the simplest radical of √12, follow these steps.

First, find out the factors of 12:

12 = 2 × 2 × 3

Therefore, the value of the root of 12 can be written as:

When you take out the square term, you get:

√12 = 2 √3

This is the radical form of √12, and you can also write it in decimal:

√12 = 2 × 1.73

√12 = ±3.46 approximately.

Simplifying radical expressions is a concept where the radical is simplified by removing the √. Since radical expressions are algebraic expressions involving radicals, it consists of roots, numbers, variables, or both. For example, the root can be a square, cube, or a n-th root. Simplifying the radical expressions implies reducing the algebraic expressions to their simplest form and eliminating the radicals from the expressions.

The simplified radical form is the process of reducing the radical expressions to their simplest form. If the radical expression is present in the denominator of an algebraic expression, you need to multiply the numerator and denominator with the accurate radical expression to simplify the equation. Let us see an example:

f(x) = √(4x^{2} y^{6})

To simplify f(x), take a look at the factors of 4x^{2} y^{6 }

f(x) = √(2 × 2 × x × x × y^{3} × y^{3}) = √(2^{2} × x^{2} × (y^{3} × y^{3})) = 2 |x| |y^{3| }

Since √x^{2 }is always negative it is |x|.

An expression having the √ sign is said to be in reduced radical form if the radicand, the number under the radical sign, does not contain any perfect squares, cubes etc. Moreover, it is an expression that includes the radical sign () and is said to be in the reduced radical form if there are no perfect squares. It can also be used to simplify a square root. You can use

a and b, √ab=√a⋅√b

here the square root of the product is similar to the product of the square roots.

Ans. The radical of a number is similar to the root of the number. It can be a square root, cube or, in general, n-th root. If n is a positive integer and is greater than 1 and a is a real number, then n√a=a1n where the left side of the equation is the radical form and the right is the exponent form. N is called the index, 'a' is the radicand, and the √ symbol is radical.

Ans. Follow the simple steps to convert an equation to a radical form:

- Use x
^{m/n}=n√x^{m }to rewrite the exponentiation as a radical. - Anything raised to 1 is the base.

Example:

Express (2x)^{1/3} in radical form.

Solution:

Rewrite the expression with a fractional exponent as a radical 3√2x. the denominator of the fraction determines the cube root, and the parenthesis indicates the exponent. Thus the answer is (2x)^{1/3} = 3√2x.

Ans. Since √7 is a prime number, it cannot be further simplified. Thus, the radical form of 7 is √7. It can be calculated using the average method of the long division method. The square root of 7 lies somewhere between the square roots of two perfect squares closer to 7.

Ans. The square root of 8 in its radical form is represented as √8 and is equal to 2√2. Thus, the answer is √8 = 2.82842712475

Or

√8 = 2.828 up to three places of decimal

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