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Evaluate expressions involving real numbers

Evaluation of an expression involving real numbers means determining its value by interpreting grouping symbols, exponents, roots, and arithmetic operations according to their structure and order of operations. This includes integers, fractions, decimals, and familiar irrational numbers, with exact forms retained when appropriate and decimal approximations used when needed; division by zero and even roots of negative numbers are not defined within the real numbers. This understanding supports substitution, algebraic manipulation, and evaluating functions.

Detailed Explanation: Evaluate expressions involving real numbers

To evaluate a real-number expression, simplify it in the correct order:

  1. Grouping symbols, such as parentheses
  2. Exponents and roots
  3. Multiplication and division from left to right
  4. Addition and subtraction from left to right

Keep exact values when possible. For example, leave 2\sqrt{2} as 2\sqrt{2} instead of rounding it.

Evaluate:

4−12(32−16)4-\frac{1}{2}\left(3^2-\sqrt{16}\right)

First, evaluate the exponent and root inside the parentheses:

32=9and16=43^2=9 \qquad\text{and}\qquad \sqrt{16}=4

Substitute these values:

4−12(9−4)4-\frac{1}{2}(9-4)

Simplify inside the parentheses:

4−12(5)4-\frac{1}{2}(5)

Multiply:

4−524-\frac{5}{2}

Rewrite 44 with denominator 22 and subtract:

82−52=32\frac{8}{2}-\frac{5}{2}=\frac{3}{2}

Therefore,

32\boxed{\frac{3}{2}}

The expression has the decimal value 1.51.5. Remember that division by zero and even roots of negative numbers are not defined as real numbers.

Learn by doing: Evaluate expressions involving real numbers

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    Order of Operations - Real Numbers, Fractions, Radicals and Exponents

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