Problem 51 - Algebra A Combined Function

Question

Evaluate. $$ 10^{2}-5^{3} $$

Step-by-Step Solution

Answer

The result of the expression is -25.

Step 1: Evaluate the Power of 10

Calculate the value of the expression \(10^2\). Recall that \(10^2\) means 10 multiplied by itself. \[10^2 = 10 \times 10 = 100\]

Step 2: Evaluate the Power of 5

Calculate the value of the expression \(5^3\). Recall that \(5^3\) means 5 multiplied by itself three times. \[5^3 = 5 \times 5 \times 5 = 125\]

Step 3: Subtract the Results

Subtract the result of \(5^3\) from the result of \(10^2\) to find the final answer. \[100 - 125 = -25\]

Key Concepts

Power of a Number

When we talk about the "power of a number," we are referring to something called "exponents." Exponents tell us how many times to multiply a number by itself. For example, in the expression \(10^2\), the number 10 is the base, and the number 2 is the exponent. This means we multiply 10 by itself: \(10 \times 10 = 100\).

Exponents make it easy to handle large numbers. Instead of writing long multiplication, like \(10 \times 10\), we simply use \(10^2\).

• A base raised to the power of 1 is the base itself. For example, \(10^1 = 10\).
• A base raised to the power of 0 is always 1, regardless of the base value. For instance, \(10^0 = 1\).

Understanding the power of a number helps in simplifying calculations and expressing large or repeated multiplication effectively.

Subtraction in Arithmetic

Subtraction is a fundamental arithmetic operation. It involves taking one number away from another. In our problem, once we evaluate the powers, we perform subtraction. For example, subtracting 125 from 100 can be written as \(100 - 125\).

Subtraction works by removing the smaller number from the larger one to find the difference, but when the larger number is being subtracted, it leads to a negative number.

• When subtracting 0 from any number, the result is the number itself. For example, \(10 - 0 = 10\).
• When a number is subtracted from itself, the result is 0. For instance, \(10 - 10 = 0\).

Subtraction helps us find the difference between quantities and is key to solving equations and understanding relative sizes.

Negative Numbers

Negative numbers are numbers less than zero. They are written with a minus sign (-) in front. In our calculation, after subtracting \(5^3\) from \(10^2\), we end up with -25, which is a negative number.

Working with negative numbers is common in areas like temperatures, elevations, and even debts. Here's how negative numbers behave:

• Adding a negative number is like subtraction. For example, adding -5 to 10 is \(10 + (-5) = 5\).
• Subtracting a negative number is like addition. For example, subtracting -5 from 10 is \(10 - (-5) = 15\).
• The product of two negative numbers is positive. For instance, \((-5) \times (-5) = 25\).

Negative numbers help us to express and work with values in contexts where decreases or reversals occur. They are vital for a complete understanding of arithmetic operations.