Questions: Simplify the expression by combining the radical terms using the indicated operation(s). 5 ∛6 + 6 ∛6 - 3 ∛7

Solution Steps

To simplify the expression, we need to combine like terms. The terms \(5 \sqrt[3]{6}\) and \(6 \sqrt[3]{6}\) are like terms because they both contain \(\sqrt[3]{6}\). We can add these coefficients together. The term \(-3 \sqrt[3]{7}\) is not like the others and remains separate.

To simplify the expression \(5 \sqrt[3]{6} + 6 \sqrt[3]{6} - 3 \sqrt[3]{7}\), we first combine the like terms \(5 \sqrt[3]{6}\) and \(6 \sqrt[3]{6}\).

\[ 5 \sqrt[3]{6} + 6 \sqrt[3]{6} = (5 + 6) \sqrt[3]{6} = 11 \sqrt[3]{6} \]

Now, we rewrite the expression by combining the result from Step 1 with the remaining term \(-3 \sqrt[3]{7}\).

\[ 11 \sqrt[3]{6} - 3 \sqrt[3]{7} \]

Next, we evaluate the cubic roots to four significant digits.

\[ \sqrt[3]{6} \approx 1.817 \] \[ \sqrt[3]{7} \approx 1.913 \]

Substitute the evaluated cubic roots back into the expression:

\[ 11 \times 1.817 - 3 \times 1.913 \]

Calculate the result:

\[ 11 \times 1.817 = 19.987 \] \[ 3 \times 1.913 = 5.739 \] \[ 19.987 - 5.739 = 14.248 \]

Final Answer