Evaluate 1/2(7+5^3)

Solution:

Step 1: Simplify the given expression.

• Step 1.1: Compute the exponentiation of $5^3$.

  $$\frac{1}{2} \times (7 + 5^3) = \frac{1}{2} \times (7 + 125)$$

• Step 1.2: Perform the addition inside the parentheses.

  $$\frac{1}{2} \times (7 + 125) = \frac{1}{2} \times 132$$

Step 2: Reduce the expression by eliminating common factors.

• Step 2.1: Extract the factor of $2$ from $132$.

  $$\frac{1}{2} \times 132 = \frac{1}{2} \times (2 \times 66)$$

• Step 2.2: Simplify by canceling out the common factor of $2$.

  $$\frac{1}{\cancel{2}} \times (\cancel{2} \times 66) = 66$$

• Step 2.3: Finalize the simplified expression.

  $$66$$

Knowledge Notes:

• Exponentiation: Raising a number to a power is a mathematical operation, symbolized by a superscript numeral. For example, $5^3$ means $5$ multiplied by itself $3$ times, which equals $125$.

• Order of Operations: When solving mathematical expressions, the order of operations must be followed. This is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). In the given problem, the exponentiation is performed before the addition.

• Simplifying Fractions: When simplifying expressions that involve fractions, look for common factors in the numerator and the denominator that can be canceled out. In this case, the common factor of $2$ in the numerator and denominator is canceled to simplify the expression.

• Arithmetic Operations: Basic arithmetic operations include addition, subtraction, multiplication, and division. These operations are used in the problem to simplify the expression step by step.

• Algebraic Manipulation: This involves rearranging and simplifying expressions using algebraic rules. In the solution, algebraic manipulation is used to factor out a $2$ and then cancel it with the $2$ in the denominator.

By understanding and applying these concepts, the original problem is solved systematically to arrive at the final answer.