Multiplication of Radicals

Multiplication of radicals

Radicals of the same order are multiplied by the formula \sqrt[n]{a}\,\sqrt[n]{b} = \sqrt[n]{ab}:

Example 1. \sqrt6 \cdot \sqrt{15} = \sqrt{90} = \sqrt{9 \cdot 10} = 3\sqrt{10}.

Example 2. 2\sqrt[3]{9} \cdot 5\sqrt[3]{6} = 10\sqrt[3]{54} = 10\sqrt[3]{27 \cdot 2} = 30\sqrt[3]2.

Example 3. \sqrt{\tfrac35} \cdot \sqrt{\tfrac{5}{12}} = \sqrt{\tfrac{15}{60}} = \sqrt{\tfrac14} = \tfrac12.

Example 4. 3\sqrt{2ab} \cdot 4\sqrt{6a^3b} = 12\sqrt{12a^4b^2} = 12 \cdot 2a^2b\sqrt3 = 24a^2b\sqrt3.

Rule

If necessary, reduce the radicals to the same order.
Multiply the coefficients for the coefficient of the result.
Multiply the radicands and write the product under the common radical sign.
Reduce the result to its simplest form.

The rule does not hold for imaginary numbers, that is, for radicals of even order with negative radicands. That case is discussed later.

How to type radicals

Square root: sqrt(12) or √12; a coefficient goes in front: 2sqrt(3) or 2√3.
Cube root: cbrt(5) or ∛5. Any other root: root(x, 4) for \sqrt[4]{x}, root(2, 6) for \sqrt[6]{2}.
Put the whole radicand in parentheses when it is more than one number or letter: sqrt(2x), cbrt(a + b), sqrt(2 - sqrt(2)).
Fractional exponents use ^ with parentheses: x^(2/3).
Letters under a radical sign stand for positive numbers, so \sqrt{x^2} = x.

Exercises

0 of 21 solved

How to type math

Use ^ for powers (a^2), / for fractions ((a+b)/2), sqrt(x) for square roots, and cbrt(x) for cube roots. You can leave out the multiplication sign: 3ab means 3 \cdot a \cdot b. Mixed numbers like 4 1/2 work too. The preview under each box shows how your answer was read.

Perform the indicated multiplications and simplify the products:

\sqrt2\cdot\sqrt8

\sqrt3\cdot\sqrt{27}

5^{1/2}\cdot 20^{1/2}

18^{1/2}\cdot 8^{1/2}

\sqrt[3]{16}\cdot\sqrt[3]4

\sqrt[3]4\cdot\sqrt[3]{12}

(100)^{1/3}\cdot(30)^{1/3}

\sqrt[4]8\cdot\sqrt[4]{32}

\sqrt{\tfrac{12}{25}}\cdot\sqrt{75}

\sqrt{\tfrac23}\cdot\sqrt{\tfrac{27}8}

\sqrt{11}\cdot\sqrt{\tfrac1{11}}

a^{1/2}\cdot(bc)^{1/2}

2\sqrt x\cdot\sqrt[2]{4x^3}

5\sqrt[3]{2a}\cdot 3\sqrt[3]{16a}

2\sqrt{rs}\cdot 7\sqrt{r^3s^3t^2}

\sqrt{\dfrac ax}\cdot\sqrt{\dfrac{4x}a}

\sqrt{75a}\cdot(45a)^{1/2}

\sqrt{2u}\cdot\sqrt{4v}\cdot\sqrt{6uv}

5\sqrt{3m}\cdot 5\sqrt{3m}

\left(3\sqrt[2]{3x}\right)^2

\sqrt[3]2\cdot\sqrt2