Lesson 9.4: Multiply Square Roots

New Concept

Master multiplying square roots using the Product Property, $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$. You'll combine simple radicals and then expand your skills to multiply binomial expressions with square roots, just like you would with polynomials.

What’s next

Now, let's see this in action. You'll walk through interactive examples and then test your skills with a series of practice cards on multiplying radicals.

Property

If $a, b$ are nonnegative real numbers, then $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$ and $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$.
To multiply radicals with coefficients, multiply the coefficients together and then the variables, just like multiplying algebraic terms.

Examples

• To simplify $\sqrt{3} \cdot \sqrt{6}$, we multiply to get $\sqrt{18}$. Then, we simplify $\sqrt{18}$ as $\sqrt{9 \cdot 2}$, which results in $3\sqrt{2}$.
• For $(5\sqrt{2})(4\sqrt{8})$, multiply the coefficients $(5 \cdot 4)$ and the radicals $(\sqrt{2} \cdot \sqrt{8})$. This gives $20\sqrt{16}$, which simplifies to $20 \cdot 4 = 80$.
• To simplify $(\sqrt{10x})(\sqrt{5x^3})$, multiply under the radical to get $\sqrt{50x^4}$. This simplifies to $\sqrt{25x^4 \cdot 2}$, which is $5x^2\sqrt{2}$.

Explanation

This property allows you to combine two separate square roots into one by multiplying the numbers under the radicals. Afterwards, always check if the new radical can be simplified by factoring out any perfect squares.

Property

If $a$ is a nonnegative real number, then $(\sqrt{a})^2 = a$.

Examples

• To simplify $(\sqrt{7})^2$, the square and square root cancel, leaving $7$.
• For $(-\sqrt{13})^2$, the expression means $(-\sqrt{13})(-\sqrt{13})$. The negatives cancel, and $\sqrt{13} \cdot \sqrt{13} = 13$.
• To simplify $(5\sqrt{3})^2$, you square both the coefficient and the radical: $5^2 \cdot (\sqrt{3})^2 = 25 \cdot 3 = 75$.

Explanation

Squaring a square root is like undoing an operation. The square and the square root cancel each other out, leaving only the number that was inside the radical. This is because $(\sqrt{a})^2$ means $\sqrt{a} \cdot \sqrt{a} = \sqrt{a^2} = a$.

Property

To multiply expressions containing square roots, use the same methods as for multiplying polynomials, such as the Distributive Property and FOIL.

Examples

• To simplify $\sqrt{5}(3 - \sqrt{10})$, distribute $\sqrt{5}$ to get $3\sqrt{5} - \sqrt{50}$. Simplify $\sqrt{50}$ to $5\sqrt{2}$, so the final answer is $3\sqrt{5} - 5\sqrt{2}$.
• For $(1 + \sqrt{5})(6 - \sqrt{5})$, use FOIL: $1 \cdot 6 + 1(-\sqrt{5}) + \sqrt{5}(6) + \sqrt{5}(-\sqrt{5})$. This is $6 - \sqrt{5} + 6\sqrt{5} - 5$, which simplifies to $1 + 5\sqrt{5}$.
• For $(2\sqrt{3} - 4)( \sqrt{3} + 5)$, use FOIL: $2\sqrt{3} \cdot \sqrt{3} + 2\sqrt{3} \cdot 5 - 4\sqrt{3} - 4 \cdot 5$. This simplifies to $2 \cdot 3 + 10\sqrt{3} - 4\sqrt{3} - 20$, which is $6 + 6\sqrt{3} - 20 = -14 + 6\sqrt{3}$.

Explanation

Treat square roots like variables when multiplying expressions. Distribute terms just as you would with polynomials, and then combine any like radicals. Always simplify any resulting radicals at the end.

Property

Use the special product formulas for binomial squares:
$(a + b)^2 = a^2 + 2ab + b^2$
$(a - b)^2 = a^2 - 2ab + b^2$
These formulas provide a shortcut for squaring binomials that contain square roots.

Examples

• To simplify $(5 + \sqrt{2})^2$, use the pattern: $5^2 + 2(5)(\sqrt{2}) + (\sqrt{2})^2$. This becomes $25 + 10\sqrt{2} + 2$, which simplifies to $27 + 10\sqrt{2}$.
• To simplify $(3 - 2\sqrt{5})^2$, use the pattern: $3^2 - 2(3)(2\sqrt{5}) + (2\sqrt{5})^2$. This is $9 - 12\sqrt{5} + 4 \cdot 5 = 9 - 12\sqrt{5} + 20$, which simplifies to $29 - 12\sqrt{5}$.
• To simplify $(2 + 3\sqrt{y})^2$, use the pattern: $2^2 + 2(2)(3\sqrt{y}) + (3\sqrt{y})^2$. This becomes $4 + 12\sqrt{y} + 9y$.

Explanation

Instead of using FOIL, you can use these shortcuts. Square the first term, square the second term, and add or subtract twice their product. This pattern simplifies squaring binomials with radicals.

Property

The Product of Conjugates formula is $(a - b)(a + b) = a^2 - b^2$.
When expressions like $(x - \sqrt{y})$ and $(x + \sqrt{y})$ are multiplied, the result contains no square roots.

Examples

• To simplify $(6 - \sqrt{5})(6 + \sqrt{5})$, use the pattern $a^2 - b^2$: $6^2 - (\sqrt{5})^2 = 36 - 5 = 31$.
• For $(3 - 4\sqrt{2})(3 + 4\sqrt{2})$, the result is $3^2 - (4\sqrt{2})^2$. This simplifies to $9 - (16 \cdot 2) = 9 - 32 = -23$.
• For $(\sqrt{7} + \sqrt{3})(\sqrt{7} - \sqrt{3})$, the result is $(\sqrt{7})^2 - (\sqrt{3})^2 = 7 - 3 = 4$.

Explanation

Multiplying conjugates, which have the same terms but opposite signs, is a special trick. The middle terms with radicals cancel out, leaving a rational number. This is because the 'Outer' and 'Inner' products are opposites.