Math Problem Solver

\(4.0071\) rounded to \(3\) significant figures is: \[ 4.01 \times 10^{0} \] In scientific notation, and \[ 4.01 \] in standard notation.

Rounding a quantity to a given number of significant digits requires identifying the most significant digits of the quantity, which can be achieved by expressing the quantity in scientific notation. A quantity \(q\) is said to be in scientific notation if \(q = m \times 10^n\), where \(1 \le \lvert m \rvert < 10\) and \(n\) is an integer. The quantities \(m\) and \(10^n\) are called the mantissa and magnitude of \(q\). Because the magnitude of the mantissa is between 1 and 10, the first digit of the mantissa is the most significant digit of the quantity \(q\), the second digit of the mantissa is the second most significant digit of \(q\), and so on.

Rounding \(4.0071\) to 3 significant figures can be achieved by expressing \(4.0071\) in scientific notation, and then rounding the mantissa to 3 significant figures. Since the magnitude of the mantissa is between 1 and 10, rounding the mantissa to 3 significant figures is equivalent to rounding it to the nearest \(10^{1 - 3} = 10^{-2}\). \[ 4.0071 \approx 4.007 \times 10^{0} \] \(4.007\) rounded to the nearest \(10^{-2}\) is \(4.01\). \(4.0071\) rounded to 3 significant figures is: \[ 4.01 \times 10^{0} \]