What is the value of 5 + 3 × 5 + 4 × 52 + 3 × 53 + 4 × 54?

Concept

In an expression containing powers, multiplication, and addition, apply the order of operations: evaluate powers first, then multiplication, and finally addition.

For a positive integer exponent n, the power an means multiplying a by itself n times. Operations at the same precedence level are then completed from left to right.

Application

1. Evaluate the powers: 52 = 5 × 5 = 25; 53 = 5 × 5 × 5 = 125; and 54 = 5 × 5 × 5 × 5 = 625.

2. Substitute these values: 5 + 3 × 5 + 4 × 25 + 3 × 125 + 4 × 625.

3. Perform the multiplications: 3 × 5 = 15, 4 × 25 = 100, 3 × 125 = 375, and 4 × 625 = 2500.

4. Add the evaluated terms: 5 + 15 + 100 + 375 + 2500 = 20 + 100 + 375 + 2500 = 120 + 375 + 2500 = 495 + 2500 = 2995.

Cross-check

The same polynomial can be evaluated by Horner’s method, which repeatedly multiplies by 5 and adds the next coefficient:

1. 4 × 5 + 3 = 23

2. 23 × 5 + 4 = 119

3. 119 × 5 + 3 = 598

4. 598 × 5 + 5 = 2995

Both methods give 2995; therefore, the value of the expression is 2995.