Problem 12 - Fundamentals of Mathematics

Question

Determine the value of each expression. \(5^{3}-\sqrt{100}+8 \cdot 2-20 \div 5\)

Step-by-Step Solution

Answer

The value of the expression is 127.

Step 1: Evaluate Exponents

First, evaluate the exponent in the expression. We have \(5^3\), which means \(5\) raised to the power of \(3\). Calculate \(5^3 = 5 \times 5 \times 5 = 125\). The expression is now \(125 - \sqrt{100} + 8 \cdot 2 - 20 \div 5\).

Step 2: Evaluate Square Roots

Now, evaluate the square root in the expression. We have \(\sqrt{100}\), which is \(10\). This simplifies our expression to \(125 - 10 + 8 \cdot 2 - 20 \div 5\).

Step 3: Evaluate Multiplication

Next, perform the multiplication operation. Multiply \(8\) by \(2\) to get \(16\). The expression becomes \(125 - 10 + 16 - 20 \div 5\).

Step 4: Evaluate Division

Now, perform the division operation. Divide \(20\) by \(5\) to get \(4\). The expression is now \(125 - 10 + 16 - 4\).

Step 5: Perform Addition and Subtraction from Left to Right

Finally, perform the remaining operations from left to right. Start with subtraction: \(125 - 10 = 115\), then add: \(115 + 16 = 131\), and finally subtract \(4\): \(131 - 4 = 127\).

Key Concepts

Exponents

Understanding exponents is essential because they are a significant part of mathematical operations. An exponent tells you how many times to multiply a number, known as the base, by itself. For instance, in the expression \(5^3\), "5" is the base, and "3" is the exponent. This particular expression means you multiply 5 three times: \(5 \times 5 \times 5\).

• The result is 125.
• Exponents can be considered a shorthand for repeated multiplication.

It's crucial to handle exponents early in a calculation, as they can significantly change a number's value.

Square Roots

Square roots can seem tricky at first, but they're easier to understand with some insight. A square root asks the question, "What number multiplied by itself gives me the original number?" In our expression, we have \(\sqrt{100}\), asking us what number squared results in 100.

• The answer is 10 because \(10 \times 10 = 100\).

Knowing how to recognize perfect squares, like 100, helps simplify expressions quickly.Studying and practicing finding square roots deepen your understanding of these mathematical fundamentals.

Multiplication and Division

When you're dealing with mathematical operations, multiplication and division are usually treated together and performed from left to right as they appear. This is illustrated in our expression, where 8 is multiplied by 2, and 20 is divided by 5. Let's break these tasks down.

• Multiply: \(8 \times 2 = 16\)
• Divide: \(20 \div 5 = 4\)

Remember, both processes are performed after handling exponents and square roots but before any addition or subtraction, according to the order of operations.

Addition and Subtraction

Addition and subtraction are the final steps in simplifying an expression, and they are performed from left to right. In the provided example, after performing the other operations, the expression boils down to \(125 - 10 + 16 - 4\).

• Subtract: \(125 - 10 = 115\)
• Add: \(115 + 16 = 131\)
• Finally, subtract: \(131 - 4 = 127\)

Pay close attention to addition and subtraction because they direct the final outcome of your calculations. Ensuring these operations are completed last helps maintain the integrity of the problem you're solving.