Roots, Radicals, Absolute Value and Indices
Roots/Radicals
If a number can be expressed as the product of two equal factors that factor is the square root of that number. √49 = 7 because 7x7 = 49. 7 is the factor that when multiplied by itself will give 49 thus making it the square root of 49. When ever we will square root a number there would be two outcomes, a positive and negative. This is because 7² = 49 and (-7)² = 49. when minus 7 will be multiplied by itself the negative sign would cancel out and will give us 49.
What if we only want the positive number and not both the outcome? For that we have the principle square root of a number which will only return the positive value of the root. The negative square root of 16 is written as -√16 not √-16. If there is no number to the left of the radical sign ( √) we assume that it is the square root. Thus for any even number on the left, if a negative number is supplied inside the radical symbol it would lead to the result being a complex number in the form a+bi | i = √-1. This comes in the parlance of imaginary numbers where if b ≠ 0 it would be an imaginary number, else it would come in the parameter of a real number.
For any odd index, the value in the radicand can be negative. A negative number when multiplied odd times will yield a negative number. ∛-64 = -4 same way that (-4)³ = -64.
Radicals/Surds
Till now we have been dealing with rational roots, some roots could be irrational such as √80 which can be simplified further but will still retain its radical sign due to having no integer square roots.
There are certain ways in which we can simplify radicals which would bring them to their simplest form:
Through this we can now simplify √80 successfully:
√80 = √8 x √10 = √4 x √2 x √10 = 2√(2 x 10) = 2√20
Thus √80 can be fully simplified into 2 √20.
√24 = √4 x √6 = 2 √6
Thus √24 can be fully simplified into 2 √6.
Rationalizing the Denominator and the Numerator
Sometimes we might have a radical in our denominator that we might want to get rid off just to make the division process simpler. The process of eliminating irrational numbers from the denominator of a fraction is called rationalizing the denominator. Since we all know that, (√80)² = 80, thus multiplying the same root into itself will yield the number under the radical without the radical.
If we have a addition or a subtraction symbol in the denominator of a fraction with a irrational denominator, we can remove the radical by multiplying it with its conjugate. The conjugate property states:
Thus if there is a radical in b it would be squared and eliminated therefore rationalizing the denominator. Below are some examples of the same:
Same way we could not want an irrational numerators in our fraction, thus we use the same process to rationalize the numerator. But in this case both the numerator and the denominator will be multiplied by the conjugate of the numerator instead the conjugate of the denominator.
In the next section we are going to talk about the properties of indices and the underlying math behind some of its conjectures.
Exponents (Indices)
Repeated multiplication of identical numbers can be written in exponential form. There could be integer exponents and there could be rational exponents both with their own respective properties. The properties of a integer exponent are:
Rational Exponents
Rational exponents are the ones which have fractions as exponents. These can be simplified by the following conjectures. Those being that if we want to equate 4^(1/4) then we must use b^m * b^n = b^(m+n) and (b^m)^n = b^mn. Thus it must follow that 4^(1/2) * 4^(1/2) = 4^(1/2 + 1/2) = 4¹
It also follows that (4^(1/2))² = 4^(1/2)*2 = 4¹. Thus we can come to define 4^(1/2) as the square root of 4. Thus we can make the conjecture that:
We can now use this same idea and develop it for exponents with a numerator other than one and a denominator. The derivation is a follows:
Now we know how to simplify or solve equations with rational exponents of all types. A few practice problems on this would solidify your understanding on this topic.
Absolute Value
The absolute value or sometimes called the modulus of a number is the distance of that number from 0 on the number line. Example, mod(-4) = 4 the same way that mod(4) = 4. Thus the mod of any number will be a positive value.
Distance of Two Points on the Real Number Line:
Given that a and b are real numbers, the distance between the points with coordinates a and b on the real number line is |b-a|, which is also the equivalent of |a-b|.
The statement the distance between k and 8 is 5.5 can be expressed algebraically as |k-8|=5/2.
Below are some examples of algebraic modulus equations and their solutions:
This sums up today’s story which includes simplification of radicals, properties of exponents and their demonstration, dealing with rational exponents and introduction to absolute value and solving algebraic modulus equations.
