Question

simplify $5sqrt3{81} + sqrt3{375}$.

Explanation:

Step1: Factor the radicands

First, factor 81 and 375 to find perfect cube factors.
For \( \sqrt[3]{81} \), we know that \( 81 = 27\times3 = 3^3\times3 \), so \( \sqrt[3]{81}=\sqrt[3]{3^3\times3}=3\sqrt[3]{3} \).
For \( \sqrt[3]{375} \), we factor 375 as \( 375 = 125\times3 = 5^3\times3 \), so \( \sqrt[3]{375}=\sqrt[3]{5^3\times3}=5\sqrt[3]{3} \).

Step2: Substitute back into the original expression

The original expression is \( 5\sqrt[3]{81}+\sqrt[3]{375} \). Substitute the simplified cube roots:
\( 5\times(3\sqrt[3]{3}) + 5\sqrt[3]{3} \)

Step3: Simplify the terms

First, calculate \( 5\times3\sqrt[3]{3}=15\sqrt[3]{3} \). Then the expression becomes:
\( 15\sqrt[3]{3}+5\sqrt[3]{3} \)

Step4: Combine like terms

Since both terms have \( \sqrt[3]{3} \), we can combine the coefficients:
\( (15 + 5)\sqrt[3]{3}=20\sqrt[3]{3} \)

Answer:

\( 20\sqrt[3]{3} \)