Question

Perform the indicated operations. Assume all variables represent positive real numbers. 2∛16 + ∛54

Accepted Answer

Rewrite each cube root expression in terms of prime factors or simpler components. For example, express $$\sqrt[3]{16}$$ and $$\sqrt[3]{54}$$ using their prime factorizations: $$16 = 2^4$ and 54 = 2$$ 	imes $$3^3. $$Use the property of cube roots that $$\sqrt[3]{a^b} = a$$^{\frac{b}$${3}} to $$simplify each term. For instance, $$\sqrt[3]{2^4$} = 2$$^{\frac{4}$${3}}$$ and $$\sqrt[3]{2 	imes 3^3$} = \sqrt[3]{2} 	imes \sqrt[3]{3^3$} = \sqrt[3]{2} 	imes 3. $$Rewrite the original expression $$2 \sqrt[3]{16} + \sqrt[3]{54}$$ using the simplified forms from step 2, so it becomes $$2 	imes 2$$^{\frac{4}$${3}} + 3 	imes \sqrt[3]{2}. $$Factor out the common cube root term $$\sqrt[3]{2} = 2$$^{\frac{1}$${3}}$$ from both terms to combine them. This means expressing each term as a multiple of $$2^{\frac{1}$${3}}$$. After factoring, combine the coefficients and write the expression as a single term involving $$\sqrt[3]{2}$$. This completes the simplification of the original expression.

Perform the indicated operations. Assume all variables represent positive real numbers. 2∛16 + ∛54

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Key Concepts

Radical Expressions and Cube Roots

Simplifying Radicals

Adding Radical Expressions