Question

Multiplication of Radicals. Multiply and simplify. $$\sqrt[3]{a^{2} b} \text { by } \sqrt[3]{2 a^{2} b^{2}} \text { and } \sqrt{3 a^{3} b^{2}}$$

Short answer

\(b\sqrt[3]{2a} \cdot \sqrt{3 a^{3} b^{2}} = b\sqrt[3]{2a^3} \cdot a\sqrt{3b^2} = ba^2\sqrt[3]{2} \cdot \sqrt{3b^2}\), which cannot be further simplified because the cube root and square root have different radicands.

Step-by-step solution

Identify Radicals

Recognize each expression as a radical. The first is a cube root of the product of variables: \(\sqrt[3]{a^{2} b}\). The second is the cube root of a product of a numerical coefficient and variables: \(\sqrt[3]{2 a^{2} b^{2}}\). The third is a simple square root of a product of variables: \(\sqrt{3 a^{3} b^{2}}\).

Multiply the Radicals

Multiply the cube roots by multiplying the insides first: \(\sqrt[3]{a^{2} b} \cdot \sqrt[3]{2 a^{2} b^{2}} = \sqrt[3]{a^{2} b \cdot 2 a^{2} b^{2}}\). The square root cannot be directly multiplied with the cube roots yet as they have different indices.

Simplify the Cube Root

Combine and simplify the terms inside the cube root: \(\sqrt[3]{a^{2} b \cdot 2 a^{2} b^{2}} = \sqrt[3]{2 a^{4} b^{3}}\). Now you can reduce the perfect cube \(b^{3}\) outside the cube root cubed and simplify \(a^{4}\) as \(a^{3} \cdot a\): \(\sqrt[3]{2 a^{4} b^{3}} = b\sqrt[3]{2 a^{3} \cdot a} = ba\sqrt[3]{2a}\).

Incorporate the Square Root

Finally, multiply the square root by the expression: \(b\sqrt[3]{2a} \cdot \sqrt{3 a^{3} b^{2}}\). You can't simplify this further since the terms inside the square root and cube root can't be combined.

Key concepts

Simplify Cube Root

When working with cube roots, the goal is to simplify the radical expression to the point where no cube within the radical can be factored out. To simplify a cube root, such as \( \sqrt[3]{a^2 b} \), you need to identify whether any factors are perfect cubes. A perfect cube is a number that can be expressed as the product of a number multiplied by itself three times, like \( 2^3 \).

In simplifying cube roots, we search for factors within the radical that are perfect cubes and separate them from those that aren't. For example, \( \sqrt[3]{8} = 2 \) because 8 is a perfect cube (\(2 \cdot 2 \cdot 2\)). However, if no perfect cube can be factored out, as with \( \sqrt[3]{a^2 b} \), the expression remains in its radical form unless it is being multiplied with another cube root with similar factors.

Simplify Square Root

Simplifying square roots revolves around finding the square root of the number or variables inside the radical. A perfect square is a number that can be expressed as some integer squared, such as \( 4 = 2^2 \).

To simplify a square root like \( \sqrt{3 a^3 b^2} \), identify the perfect squares within the radical. Here, \( b^2 \) is a perfect square and can be simplified, so \( \sqrt{3 a^3 b^2} = b\sqrt{3a^3} \). The remaining part under the square root that cannot be simplified (as \(3a^3\) is not a perfect square when considering the \(a\)'s exponent) stays inside the radical.

Radical Expressions Multiplication

To multiply radical expressions, particularly different types of roots like square roots and cube roots, one must initially consider if the indices (the small number outside the radical that indicates the type of root) are the same. When you have identical indices (as with cube roots), you can multiply the insides directly, such as in this expression: \(\sqrt[3]{a^2 b} \cdot \sqrt[3]{2 a^2 b^2} = \sqrt[3]{a^2 b \cdot 2 a^2 b^2}\).

However, you cannot directly multiply a cube root with a square root because their indices are different – cube roots are related to the third power, while square roots are related to the second power. You may simplify them separately first, then multiply the simplified results, if possible. Any simplification involves reducing expressions to contain only non-perfect squares or cubes within the radical.

Mathematical Simplification

Mathematical simplification includes reducing expressions to their simplest form by performing all possible arithmetic operations and factoring. In the context of radicals, simplification involves removing all perfect powers from under the radical sign by expressing them as their corresponding integer outside of the radical.

For example, during the simplification process of \(\sqrt[3]{2 a^4 b^3}\), we identify \(b^3\) as a perfect cube, which can be brought out of the cube root as \(b\), and \(a^4\) as \(a^3 \cdot a\), which simplifies to \(a \sqrt[3]{2a}\) outside the radical. To achieve robust understanding and proper simplification, recognizing perfect squares, cubes, and higher powers is essential, as well as being capable of multiplying or adding expressions under different radicals separately.