## How to Simplify Radical Expressions

Radical expressions are simplified by taking them down to their simplest form and, if feasible, altogether deleting the radical. When a radical expression appears in the denominator of an algebraic expression, the radical expression is simplified by multiplying the numerator and denominator by the appropriate radical expression (for example, conjugate in the case of a binomial, the same radical in the case of a monomial).

### Simplifying Radical Expressions with Square Root

Let's look at an example of simplifying radical expression with square root. Take $$\sqrt{2700}$$ as an example. We shall reduce this radical expression to its simplest form until it cannot be reduced any further.

Step1: Factorize the number under the radical.
$$2700 \ = \ 3 \times 3 \times 3 \times 2 \times 2 \times 5 \times 5$$

Step2: The number under the radical should be written as the product of its factors as powers of $$2$$.
$$2700 \ = \ 3^2 \times 3 \times 2^2 \times 5^2$$

Step3: Factors that have the power two should be written outside the radical.
$$\sqrt{2700} \ = \ 3 \times 2 \times 5 \ \sqrt{3}$$

Step4: Simplify the radical to no more simplification is possible.
$$\sqrt{2700} \ = \ 3 \times 2 \times 5 \ \sqrt{3} \ = \ 30 \ \sqrt{3}$$

Therefore, the radical expression $$\sqrt{2700}$$ has been simplified to the its simplest form $$30 \ \sqrt{3}$$.

### Simplifying Radical Expressions with Variables

Similar to how radical expressions with numbers are simplified, radical expressions with variables are also simplified. Along with the numbers, we factor the variables. For a better understanding, let's look at an example of how to simplify radical expressions with variables.

Example: Simplify $$\sqrt{175 \ z^2 \ y^3 \ x^4}$$

Step1: Expand the variables and factorize the number under the radical.
$$\sqrt{175 \ z^2 \ y^3 \ x^4} \ = \ \sqrt{7 \times 5 \times 5 \times z \times z \times y \times y \times y \times x \times x \times x \times x \times x}$$

Step2: Pair the factors that are the same together.
$$\sqrt{7 \times 5^2 \times z^2 \times y^2 \times y \times x^2 \times x^2} \ = \ \sqrt{7 \times 5^2 \times z^2 \times y^2 \times y \times (x^2)^2}$$

Step3: Radical's components that have the power two should be written outside the radical. Notice that $$\sqrt{x^2}$$ is always positive. Therefore, to make the value non-negative, use the absolute value sign.
$$\sqrt{175 \ z^2 \ y^3 \ x^4} \ = \ \sqrt{7 \times 5^2 \times z^2 \times y^2 \times y \times x^2 \times x^2} \ =\ 5 \times |z| \times |y| \times |x^2| \sqrt{7y}$$

Step4: Simplify the radical to no more simplification is possible.
$$\sqrt{175 \ z^2 \ y^3 \ x^4} \ = \ 5 \ |z| \ |y| \ x^2 \ \sqrt{7y}$$

### Rules for Simplifying Radical Expressions

• $$\sqrt{a \ b} \ = \ \sqrt{a} \ \sqrt{b}$$
• $$\sqrt{\frac{a}{b} \ } \ = \ \frac{\sqrt{a}}{\sqrt{b}}$$ , $$b \ ≠ \ 0$$
• $$\sqrt{a} \ + \ \sqrt{b} \ ≠ \ \sqrt{a \ + \ b}$$
• $$\sqrt{a} \ - \ \sqrt{b} \ ≠ \ \sqrt{a \ - \ b}$$

### Exercises for Simplifying Radical Expressions

1) Simplify: $$\sqrt{25 \ n^2}$$

2) Simplify: $$\sqrt{27 \ n^3 \ x^2}$$

3) Simplify: $$\sqrt{576 \ z^7 \ y^5}$$

4) Simplify: $$\sqrt{1225 \ p^5 \ q^8}$$

5) Simplify: $$\sqrt{108 \ m^6 \ n^9}$$

6) Simplify: $$\sqrt{48 \ r^3 \ t^7}$$

7) Simplify: $$\sqrt{486 \ p^{11} \ q^5}$$

8) Simplify: $$\sqrt{450 \ p^{14} \ q^9}$$

9) Simplify: $$\sqrt{98 \ r^9 \ t^3}$$

10) Simplify: $$\sqrt{200 \ m^{15} \ n^{19}}$$

1) Simplify: $$\sqrt{25 \ n^2}$$

$$\color{red}{\sqrt{25 \ n^2} \ = \ \sqrt{5^2 \times n^2} \ = \ 5n}$$

2) Simplify: $$\sqrt{27 \ n^3 \ x^2}$$

$$\color{red}{\sqrt{27 \ n^3 \ x^2} \ = \ \sqrt{3^3 \times n^2 \times n \times x^2} \ = \ 3 \ n \ x\sqrt{3n}}$$

3) Simplify: $$\sqrt{576 \ z^7 \ y^5}$$

$$\color{red}{\sqrt{576 \ z^7 \ y^5} \ = \ \sqrt{4^3 \times 3^2 \times z^{2 \times 3} \times z \times y^{2 \times 2} \times y} \ = \ 4 \times 3 \ z^3 \ y^2\sqrt{4 \ z \ y} \ = \ 12 \ z^3 \ y^2\sqrt{4 \ z \ y}}$$

4) Simplify: $$\sqrt{1225 \ p^5 \ q^8}$$

$$\color{red}{\sqrt{1225 \ p^5 \ q^8} \ = \ \sqrt{5^2 \times 7^2 \times p^{2 \times 2} \times p \times q^{2 \times 4}} \ = \ 5 \times 7 \ p \ q^4\sqrt{p} \ = \ 21pq^4\sqrt{p}}$$

5) Simplify: $$\sqrt{108 \ m^6 \ n^9}$$

$$\color{red}{\sqrt{108 \ m^6 \ n^9} \ = \ \sqrt{3^3 \times 2^2 \times m^{2 \times 3} \times n \times n^{2 \times 4}} \ = \ 3 \times 2 \ m^3 \ n^4\sqrt{3n} \ = \ 6 \ z^3 \ y^2\sqrt{3n}}$$

6) Simplify: $$\sqrt{48 \ r^3 \ t^7}$$

$$\color{red}{\sqrt{48 \ r^3 \ t^7} \ = \ \sqrt{4^2 \times 3 \times r^2 \times r \times t \times t^{2 \times 3}} \ = \ 4 \ r \ t^3\sqrt{3 \ t \ r}}$$

7) Simplify: $$\sqrt{486 \ p^{11} \ q^5}$$

$$\color{red}{\sqrt{486 \ p^{11} \ q^5} \ = \ \sqrt{3^5 \times 2 \times p^{2 \times 5} \times p \times q^{2 \times 2} \times q} \ = \ 3^2 \ p^5 \ q^2\sqrt{3 \times 2 \ p \ q} \ = \ 9 \ p^5 \ q^2\sqrt{6 \ p \ q}}$$

8) Simplify: $$\sqrt{450 \ p^{14} \ q^9}$$

$$\color{red}{\sqrt{450 \ p^{14} \ q^9} \ = \ \sqrt{5^2 \times 3^2 \times 2 \times p^{2 \times 7} \times q^{2 \times 4} \times q} \ = \ 5 \times 3 \ p^7 \ q^4\sqrt{2 \ q} \ = \ 15 \ p^7 \ q^4\sqrt{2 \ q}}$$

9) Simplify: $$\sqrt{98 \ r^9 \ t^3}$$

$$\color{red}{\sqrt{98 \ r^9 \ t^3} \ = \ \sqrt{7^2 \times 2 \times r^{2 \times 4} \times r \times t^2 \times t} \ = \ 7 \ r^4 \ t\sqrt{2 \ r \ t}}$$

10) Simplify: $$\sqrt{200 \ m^{15} \ n^{19}}$$

$$\color{red}{\sqrt{200 \ m^{15} \ n^{19}} \ = \ \sqrt{5^2 \times 2^3 \times m^{2 \times 7} \times m \times n^{2 \times 9} \times n} \ =}$$ $$\color{red}{ \ 5 \times 2 \ m^7 \ n^9\sqrt{2 \ m \ n} \ = \ 10 \ m^7 \ n^9\sqrt{2 \ m \ n}}$$

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