Question

³√(c³ a¹¹ t¹²)

Explanation:

Step1: Recall the property of cube roots

For any real number \( x \) and positive integer \( n \), \( \sqrt[3]{x^m}=x^{\frac{m}{3}} \) when \( m \) is a multiple of 3, and we can apply this property to each term inside the cube root separately.

Step2: Simplify each term

• For the term \( c^{3} \): Using the property \( \sqrt[3]{x^m}=x^{\frac{m}{3}} \), we have \( \sqrt[3]{c^{3}} = c^{\frac{3}{3}}=c \).
• For the term \( a^{11} \): We can write \( a^{11}=a^{9 + 2}=a^{9}\times a^{2} \). Then \( \sqrt[3]{a^{11}}=\sqrt[3]{a^{9}\times a^{2}}=\sqrt[3]{a^{9}}\times\sqrt[3]{a^{2}} \). Since \( \sqrt[3]{a^{9}} = a^{\frac{9}{3}}=a^{3} \), so \( \sqrt[3]{a^{11}}=a^{3}\sqrt[3]{a^{2}} \).
• For the term \( t^{12} \): Using the property \( \sqrt[3]{x^m}=x^{\frac{m}{3}} \), we have \( \sqrt[3]{t^{12}}=t^{\frac{12}{3}} = t^{4} \).

Step3: Combine the simplified terms

Multiplying the simplified terms together, we get \( \sqrt[3]{c^{3}a^{11}t^{12}}=c\times a^{3}\sqrt[3]{a^{2}}\times t^{4}=a^{3}c t^{4}\sqrt[3]{a^{2}} \). Alternatively, we can express \( a^{11} \) as \( a^{3\times3 + 2} \) and use the property of exponents for radicals directly: \( \sqrt[3]{c^{3}a^{11}t^{12}}=c^{3/3}a^{11/3}t^{12/3}=c a^{\frac{9 + 2}{3}}t^{4}=c a^{3+\frac{2}{3}}t^{4}=a^{3}c t^{4}a^{\frac{2}{3}}=a^{3}c t^{4}\sqrt[3]{a^{2}} \).

Answer:

\( a^{3}c t^{4}\sqrt[3]{a^{2}} \) (or equivalent form \( c a^{3} t^{4} \sqrt[3]{a^{2}} \))