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Reference

Properties of Square Roots

Rule

Product Property of Square Roots

Given two non-negative numbers a and b, the square root of their product equals the product of the square root of each number.


sqrt(ab) = sqrt(a)* sqrt(b), for a≥ 0 and b≥ 0

Proof
Let x, y, and z be three non-negative numbers such that x=sqrt(a), y=sqrt(b), and z = sqrt(ab). By the definition of a square root, each of these numbers squared is equal to its corresponding radicand. x^2 = a & (I) y^2 = b & (II) z^2 = ab & (III) Next, multiply Equation (I) by y^2. x^2=& a ⇓ x^2 y^2=& a y^2 Now, substitute Equations (II) and (III) into this equation.
x^2 y^2 = a y^2
▼
Substitute values and simplify
Substitute

y^2= b

x^2 y^2 = a b
Substitute

ab= z^2

x^2 y^2 = z^2
ProdPowII

a^m b^m = (a b)^m

(xy)^2 = z^2
RearrangeEqn

Rearrange equation

z^2 = (xy)^2
Since x, y, and z are non-negative, the final equation implies that z=xy. z^2=(xy)^2 ⇒ z=xy The last step is substituting z=sqrt(ab), x=sqrt(a), and y=sqrt(b) into this equation. z=xy ⇔ sqrt(ab)=sqrt(a)* sqrt(b)
Rule

Quotient Property of Square Roots

Let a be a non-negative number and b be a positive number. The square root of the quotient ab equals the quotient of the square roots of a and b.


sqrt(a/b) = sqrt(a)/sqrt(b), for a≥ 0, b> 0

Proof
Let x, y, and z be non-negative numbers such that x=sqrt(a), y=sqrt(b), and z = sqrt(ab). By the definition of a square root, each of these numbers squared is equal to its corresponding radicand. x^2 = a & (I) y^2 = b & (II) z^2 = a/b & (III) Since b is a positive number, y^2 is also positive. Therefore, Equation (I) can be divided by y^2. x^2=& a ⇓ x^2/y^2=& a/y^2 Now, substitute Equations (II) and (III) into this equation.
x^2/y^2 = a/y^2
▼
Substitute values and simplify
Substitute

y^2= b

x^2/y^2 = a/b
Substitute

a/b= z^2

x^2/y^2 = z^2

a^m/b^m=(a/b)^m

(x/y)^2 = z^2
RearrangeEqn

Rearrange equation

z^2=(x/y)^2
Since x, y, and z are non-negative, the final equation implies that z= xy. z^2=(x/y)^2 ⇒ z=x/y The last step is substituting z=sqrt(ab), x=sqrt(a), and y=sqrt(b) into this equation. z=x/y ⇔ sqrt(a/b)=sqrt(a)/sqrt(b)
Exercises

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