Question

1. describe and correct the error in simplifying the expression.

× (sqrt3{16} = 4)

Explanation:

Step1: Recall cube root definition

A cube root of a number \( x \) is a number \( y \) such that \( y^3 = x \). Let's check \( 4^3 \).
\( 4^3 = 4\times4\times4 = 64 \), not 16. So \( \sqrt[3]{16}
eq4 \).

Step2: Simplify \( \sqrt[3]{16} \)

Factor 16: \( 16 = 8\times2 \), and \( 8 = 2^3 \). So \( \sqrt[3]{16}=\sqrt[3]{8\times2} \).
Using the property \( \sqrt[3]{ab}=\sqrt[3]{a}\times\sqrt[3]{b} \) (for real numbers \( a,b \)), we get \( \sqrt[3]{8\times2}=\sqrt[3]{8}\times\sqrt[3]{2} \).
Since \( \sqrt[3]{8} = 2 \) (because \( 2^3 = 8 \)), then \( \sqrt[3]{16}=2\sqrt[3]{2} \).

Answer:

The error is that \( 4^3 = 64
eq16 \), so \( \sqrt[3]{16}
eq4 \). The correct simplification is \( \sqrt[3]{16}=2\sqrt[3]{2} \).