Problem 96 - Algebra A Combined Function

Question

Simplify each expression. $$ 7^{2}-7^{0} $$

Step-by-Step Solution

Answer

The expression simplifies to 48.

Step 1: Recognize Powers

Identify the powers of 7 in the given expression. You have the terms: \(7^2\) and \(7^0\). The expression is \(7^2 - 7^0\).

Step 2: Evaluate Powers

Calculate each power separately: \(7^2 = 49\) because \(7 \times 7 = 49\). Additionally, any number to the power of zero is 1, therefore \(7^0 = 1\).

Step 3: Substitute and Simplify

Substitute the calculated values back into the expression: \(49 - 1\). Simplify this expression to get 48.

Key Concepts

Powers of Numbers

In mathematics, when we talk about powers of numbers, we are referring to how many times a number, called the base, is multiplied by itself. For instance, in the power expression \(7^2\), the number 7 is the base, and 2 is the exponent.

This means 7 is multiplied by itself once, which is equivalent to saying \(7 \times 7\). Calculating this gives us 49, showing the usefulness of understanding powers as a way to simplify multiplication.

Similarly, any number raised to the power of 0 equals 1. This concept is rooted in the property that any non-zero number to the zero power will always equal 1. For example, \(7^0 = 1\). Remembering these rules can make working with powers straightforward and easy.

Evaluating Exponents

Evaluating exponents, or calculating powers, requires knowing the base and the exponent. The exponent tells us how many times to multiply the base by itself. Here's how you can evaluate an exponent:

• Identify the base and the exponent, for example, \(7^2\).
• Calculate by multiplying the base by itself as many times as the exponent indicates: \(7 \times 7 = 49\).

When the exponent is 0, it indicates a special rule. Regardless of what the base is, as long as it's not zero, raising it to the power of zero equals 1. That's why \(7^0 = 1\). Understanding this becomes especially helpful when simplifying expressions where exponents are involved.

Subtraction in Algebra

Subtraction in algebra is similar to basic arithmetic subtraction, but it often involves variables and expressions with different terms. To apply subtraction properly:

• Find the difference between two numbers or terms as in \(49 - 1\).
• Subtract directly when you have simplified terms, which results in a clean and direct calculation: here it equals 48.

Remember that careful organization and simplification of terms are crucial before performing subtraction. In our example, we first evaluated the powers of numbers, then subtracted the results to find the answer. This step-by-step approach helps prevent errors and ensures clarity.