Questions: Simplify the expression. ∛128 / 2∛2 Simplify the expression and eliminate any negative exponent(s). Assume that all letters der (a) x^(2 / 3) x^(7 / 3) x^3

Solution Steps

1. For the first expression \(\frac{\sqrt[3]{128}}{2 \sqrt[3]{2}}\), we need to simplify the cube roots and then divide the results. The cube root of 128 can be simplified by expressing 128 as a power of 2, and similarly for the cube root of 2.

2. For the second expression \(x^{2/3} x^{7/3}\), we apply the property of exponents that states when multiplying like bases, we add the exponents.

To simplify the expression \(\frac{\sqrt[3]{128}}{2 \sqrt[3]{2}}\), we first express 128 as a power of 2: \(128 = 2^7\). Therefore, \(\sqrt[3]{128} = (2^7)^{1/3} = 2^{7/3}\). Similarly, \(\sqrt[3]{2} = 2^{1/3}\).

The expression becomes \(\frac{2^{7/3}}{2 \cdot 2^{1/3}}\). Simplifying the denominator, we have \(2 \cdot 2^{1/3} = 2^{1 + 1/3} = 2^{4/3}\). Thus, the expression simplifies to:

\[ \frac{2^{7/3}}{2^{4/3}} = 2^{(7/3) - (4/3)} = 2^{3/3} = 2^1 = 2 \]

For the expression \(x^{2/3} x^{7/3}\), we use the property of exponents that states when multiplying like bases, we add the exponents:

\[ x^{2/3} \cdot x^{7/3} = x^{(2/3) + (7/3)} = x^{9/3} = x^3 \]

Final Answer