# Simplifying square roots

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This solution deals with simplifying square roots.

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## Step by Step Solution

#### Simplify : sqrt(275t^{8}pq^{7})

## Step 1 :

#### Simplify the Integer part of the SQRT

Factor 275 into its prime factors

275 = 5^{2} • 11

To simplify a square root, we extract factors which are squares, i.e., factors that are raised to an even exponent.

Factors which will be extracted are :

25 = 5^{2}

Factors which will remain inside the root are :

11 = 11

To complete this part of the simplification we take the squre root of the factors which are to be extracted. We do this by dividing their exponents by 2 :

5 = 5

At the end of this step the partly simplified SQRT looks like this:

5 • sqrt (11t^{8}pq^{7})

## Step 2 :

#### Simplify the Variable part of the SQRT

Rules for simplifing variables which may be raised to a power:

(1) variables with no exponent stay inside the radical

(2) variables raised to power 1 or (-1) stay inside the radical

(3) variables raised to an even exponent: Half the exponent taken out, nothing remains inside the radical. examples:

(3.1) sqrt(x^{8})=x^{4}

(3.2) sqrt(x^{-6})=x^{-3}

(4) variables raised to an odd exponent which is >2 or <(-2) , examples:

(4.1) sqrt(x^{5})=x^{2}•sqrt(x)

(4.2) sqrt(x^{-7})=x^{-3}•sqrt(x^{-1})

Applying these rules to our case we find out that

SQRT(t^{8}pq^{7}) = t^{4}q^{3} • SQRT(pq)

#### Combine both simplifications

sqrt (275t^{8}pq^{7}) =

5 t^{4}q^{3} • sqrt(11pq)

## Simplified Root :

5 t^{4}q

^{3}• sqrt(11pq)