Approximate the real-number expression. Express the answer...

Approximate the real-number expression. Express the answer in sclentific notation accurate to four significant figures.
(a) $\sqrt{\left|3.45-1.2 \times 10^{4}\right|+10^{3}}$
(b) $\left(1.791 \times 10^{2}\right) \times\left(9.84 \times 10^{3}\right)$

Key Concepts

Square Root

The square root operation is the inverse of squaring a number; it refers to a value that, when multiplied by itself, produces the original number. This concept is essential in various mathematical procedures, including solving equations and simplifying expressions that involve quadratic forms.

Absolute Value

The absolute value of a number is the non-negative value of that number without regard to its sign. In mathematical expressions, using the absolute value ensures that distance or magnitude is considered regardless of the direction, which can be important when the sign might otherwise affect the outcome of operations like taking a square root.

Arithmetic with Exponents

Arithmetic with exponents involves rules that govern operations on powers of ten and other numbers. This includes the multiplication and division of numbers in scientific notation, where the coefficients are manipulated and the exponents are added or subtracted according to exponent laws, facilitating simpler handling of very large or small numbers.

Scientific Notation

Scientific notation is a method of expressing very large or very small numbers in the form of a product of a number between 1 and 10 and a power of ten. This format simplifies arithmetic operations, comparisons, and helps manage significant digits more effectively by clearly separating the scale (power of ten) from the significant figures.

Significant Figures

Significant figures refer to the digits in a number that carry meaningful information about its precision. Calculations are often rounded to a specified number of significant figures to reflect the precision of measured or computed values, which is crucial in ensuring accuracy is maintained throughout mathematical and scientific computations.

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