Problem 65 - Algebra A Combined Function

Question

Place in the appropriate space to make each a true statement. $$ (-3)^{2}-3^{2} $$

Step-by-Step Solution

Answer

The expression evaluates to 0.

Step 1: Evaluate the First Exponentiation

Begin by evaluating the expression \((-3)^2\). The base here is \(-3\) and when raised to the power of 2, it becomes \((-3) \times (-3)\). Since a negative number multiplied by a negative number results in a positive number, the result is 9.

Step 2: Evaluate the Second Exponentiation

Next, evaluate the expression \(3^2\). The base here is 3 and when raised to the power of 2, it becomes \(3 \times 3\). This results in 9.

Step 3: Subtract the Results

Now, subtract the result from Step 2 from the result of Step 1. This means calculating \(9 - 9\), which equals 0.

Key Concepts

Negative Numbers

Negative numbers are numbers that are less than zero. They are usually represented with a minus sign. A negative number like -3 is located to the left of zero on the number line. Handling negative numbers can be tricky, but with a little practice, it becomes easier.

• When you multiply two negative numbers, the result is positive. For example, \((-3)\times(-3) = 9\).
• When you multiply a negative number by a positive number, the result is negative. For example, \( (-3) \times 3 = -9 \).

Understanding these rules helps simplify challenges when working with various arithmetic operations involving negative numbers. Practicing with examples will help solidify your understanding of these concepts. Confidently managing negative numbers will significantly improve your mathematical skills.

Power of a Number

The power of a number involves using exponents. An exponent tells you how many times to multiply the base number by itself. When you see expressions like \((3)^2\), this is read as "3 squared." The base number is 3, and the exponent, which is 2, indicates you should multiply 3 by itself: \(3 \times 3 = 9\).

• A base raised to the power of 1 remains the same, e.g., \(3^1 = 3\).
• A base raised to the power of 0 is always 1, e.g., \(3^0 = 1\).

When dealing with negative bases, as in \((-3)^2\), square the entire expression, not just the 3, which is why \((-3)\times(-3) = 9\). Remember, parentheses help convey the correct part of the expression you need to solve.

Subtraction of Integers

Subtraction of integers, where integers are numbers without fractional parts, involves finding the difference between numbers. You might recall the phrase "take away," which is foundational to understanding subtraction.

• When you subtract a larger number from a smaller one, the result is negative, e.g., \(3 - 5 = -2\).
• If you subtract two equal numbers, the result is zero, e.g., \(9 - 9 = 0\).

In our example, subtracting the two equal numbers obtained through exponentiation, \(9 - 9 = 0\), shows how subtraction functions when numbers have equal value. Maintaining awareness of whether you are subtracting larger from smaller numbers will keep calculations accurate. It is always helpful to visualize integer subtraction on a number line or through simple drawings to grasp the concept more physically.