Radicals

• Cube root of 27: $\sqrt[3]{27} = 27^{\frac{1}{3}} = 3$
• Fourth root of 16: $\sqrt[4]{16} = 16^{\frac{1}{4}} = 2$
• Fifth root of 32: $\sqrt[5]{32} = 32^{\frac{1}{5}} = 2$

You can simplify radicals by using prime factorization. Let's see an example:

Find $\sqrt{75}$:

• Prime factorization of 75: $75 = 3 \times 5 \times 5$
• Since we are taking the square root, group the factors into pairs: $(5 \times 5)$.
• Taking one factor from each group: $5$.
• Thus, $\sqrt{75} = 5\sqrt{3}$.

Different operations like addition, multiplication, and division can be done on alike radicals, just like you perform operations on algebraic terms.

Example 1: Simplify the following radicals

Solution:

Simplify $\sqrt{50}$ and $\sqrt{18}$ separately:

Now, add the simplified radicals:

Example 2: Simplify the following expression

Solution:

Simplify $\sqrt{72}$ and $\sqrt{32}$ separately:

Now, subtract the simplified radicals:

If you are given a fraction expression with a radical value in the denominator, you can rationalize the denominator. This process involves expressing the fraction in such a way that there are no radical values in the denominator.

Example 1: Rationalize the denominator of the following expression

Solution:

Multiply both the numerator and denominator by $\sqrt{2}$ to rationalize the denominator:

Example 2: Rationalize the denominator for the following expression

Solution:

Multiply both the numerator and denominator by the conjugate of the denominator, $\sqrt{5} - 2$:

So the rationalized form is $3(\sqrt{5} - 2)$.

Besides square roots and cube roots, there are higher roots such as the fourth root, fifth root, etc. Let's look at a few examples:

• Fourth root of 81: $\sqrt[4]{81} = 81^{\frac{1}{4}} = 3$
• Fifth root of 32: $\sqrt[5]{32} = 32^{\frac{1}{5}} = 2$
• Sixth root of 729: $\sqrt[6]{729} = 729^{\frac{1}{6}} = 3$

Square roots and cube roots of numbers can also be obtained from mathematical tables. For example:

• Square root of 25 = 5
• Cube root of 64 = 4
• Fourth root of 81 = 3

Radicals are not just limited to pure mathematics; they have practical applications in real life. Here are a few examples:

• Engineering: Finding the length of sides in right-angled triangles using Pythagoras' theorem, $a^2 + b^2 = c^2$, involves square roots.
• Physics: Square roots are used in calculating the velocity and acceleration of objects.
• Finance: Square roots are used in the calculation of the standard deviation in statistics, which is used to measure risk in investments.