Grade 10 questions on how to multiply expressions with radicals are presented below. Try to solve each expression, then click the arrow to view the step-by-step solutions.
The multiplication formula for radicals with equal indices is given by:
Review the following examples to see how the product rule is applied:
Use the multiplication formula to simplify the following expressions:
Simplify: $4\sqrt{2} \cdot 7\sqrt{32}$
$$4\sqrt{2} \cdot 7\sqrt{32} = (4 \cdot 7)\sqrt{2 \cdot 32}$$
$$= 28\sqrt{64} = 28 \cdot 8 = 224$$
Simplify: $6\sqrt{x} \cdot \frac{2}{3}\sqrt{x}$
$$6\sqrt{x} \cdot \frac{2}{3}\sqrt{x} = \left(6 \cdot \frac{2}{3}\right)\sqrt{x \cdot x}$$
$$= 4\sqrt{x^2} = 4|x| = 4x \quad (\text{assuming } x \geq 0)$$
Simplify: $2\sqrt[3]{\frac{1}{32}} \cdot \sqrt[3]{128} \cdot \sqrt[3]{16}$
$$2\sqrt[3]{\frac{1}{32}} \cdot \sqrt[3]{128} \cdot \sqrt[3]{16} = 2\sqrt[3]{\frac{1}{32} \cdot 128 \cdot 16}$$
$$= 2\sqrt[3]{64} = 2 \cdot 4 = 8$$
Simplify: $-8\sqrt[5]{x^2} \cdot 2\sqrt[5]{x^3}$
$$-8\sqrt[5]{x^2} \cdot 2\sqrt[5]{x^3} = (-8 \cdot 2)\sqrt[5]{x^2 \cdot x^3}$$
$$= -16\sqrt[5]{x^5} = -16x$$
Simplify: $\sqrt{(x-3)} \cdot \sqrt{(x-3)}$
$$\sqrt{(x-3)} \cdot \sqrt{(x-3)} = \sqrt{(x-3)(x-3)} = \sqrt{(x-3)^2}$$
$$= |x-3| = x-3$$
(Note: Because $\sqrt{(x-3)}$ is a real number, $(x-3) \geq 0$, which allows us to simplify the absolute value $|x-3|$ directly to $x-3$.)
Simplify: $\sqrt[8]{x} \cdot 5\sqrt[8]{x^4} \cdot 2\sqrt[8]{x^3}$
$$\sqrt[8]{x} \cdot 5\sqrt[8]{x^4} \cdot 2\sqrt[8]{x^3} = (1 \cdot 5 \cdot 2)\sqrt[8]{x \cdot x^4 \cdot x^3}$$
$$= 10\sqrt[8]{x^8} = 10|x| = 10x \quad (\text{assuming } x \geq 0)$$