Problem 58 - Elementary Algebra

Question

For exercises 1-80, evaluate. $$ 5+8^{2} \div\left(20-2^{2}\right) $$

Step-by-Step Solution

Answer

9

Step 1: Evaluate the exponent

Calculate the value of the exponent first. In this case, evaluate \( 8^2 \)which equals 64.

Step 2: Evaluate the expression inside the parentheses

Next, solve the expression inside the parentheses, \( 20 - 2^2 \).First, evaluate the exponent \( 2^2 = 4 \),then subtract: \( 20 - 4 = 16 \).

Step 3: Perform the division

Now, divide the result of the exponent by the result of the subtraction, \( 64 \div 16 \).This equals 4.

Step 4: Final Addition

Lastly, add the result of the division to 5: \( 5 + 4 = 9 \).

Key Concepts

Evaluating Exponents

Evaluating exponents is an important step in many mathematical problems. It involves raising a number to the power of an exponent. In our example, we have to evaluate \(8^2\). This means 8 is multiplied by itself \(2\) times: \(8 \times 8 = 64\). Always perform exponentiation before other operations according to the order of operations (PEMDAS/BODMAS). This ensures calculations are accurate.

Division

In mathematics, division is the operation of distributing a number into a specified number of equal parts. In our example, 64 (from the evaluated exponent) is divided by the result of the expression inside the parentheses, which is 16. So, we perform \(64 \div 16 = 4\). Dividing correctly involves ensuring that the number is evenly split into the divisor, and it comes after evaluating expressions inside parentheses and exponents.

Subtraction

Subtraction is the operation of removing objects from a collection. In our step-by-step example, parentheses contain the expression \(20 - 2^2\). First, we evaluate the exponent: \(2^2 = 4\), then we subtract \(4\) from \(20\), which gives us \(16\). It is crucial to first address any exponents within parentheses before performing the subtraction to maintain accuracy.

Addition

Addition is the process of finding the total, or sum, by combining two or more numbers. In the final step of our exercise, we add \(5\) to the result of our division \(4\). So we calculate \(5 + 4 = 9\). Addition is usually performed last according to the order of operations if no parentheses alter this order, ensuring all other operations are completed first.