Problem 56 - Prealgebra

Question

Simplify each expression $$(2+6)^{2}+(-5+6)^{2}$$

Step-by-Step Solution

Answer

The simplified expression is 65.

Step 1: Simplify Inside the Parentheses

First, simplify the expressions inside the parentheses. For \( (2+6) \), add 2 and 6 to get 8. For \( (-5+6) \), add -5 and 6 to get 1.

Step 2: Square Each Result

Now, square the results from Step 1. Calculate \( 8^2 \) which is 64 and \( 1^2 \) which is 1.

Step 3: Add the Squared Results

Finally, add the squared results from Step 2 together. \( 64 + 1 = 65 \).

Key Concepts

Understanding Parentheses

Parentheses play a crucial role in many mathematical expressions. They signal that the operations within them should be performed first before handling other parts of the expression.
This helps keep the calculation organized and ensures you get the correct answer. When you see an expression with parentheses, like \(2+6\), your task is to solve whatever is inside them first.

In our original exercise, you notice two separate sets of parentheses: \(2+6\) and \(-5+6\). Always solve each one entirely before moving to the next mathematical operation.
This simplifies our work and clears up our way for future operations, such as squaring the result.
Parents are alike in how they sort out the chores, and parentheses help sort out the math tasks!

Exploring Exponents

Exponents are a way to express repeated multiplication of the same number. They are displayed as a small number, called the exponent, placed above and to the right of the base number.
For example, in \(8^2\), 8 is the base, and 2 is the exponent. This means that 8 is multiplied by itself, leading to \(8 \times 8\).

In our exercise, exponents are used to square the results inside the parentheses.

• For \(8^2\), it equals 64 because 8 \times 8 = 64.
• For \(1^2\), it equals 1, since 1 \times 1 = 1.

Exponents are powerful in showing how quickly numbers can grow with multiplication!

Mastering Addition in Expressions

Addition is one of the basic arithmetic operations in math. It's often the first operation you'll do inside parentheses and throughout an entire expression.
In our exercise, addition was used in two main steps: simplifying inside parentheses and adding the final results.

• Inside the parentheses: \(2+6 = 8\) and \(-5+6 = 1\).
• After handling the exponents, we added the results together: \(64 + 1 = 65\).

Addition combines numbers to find their total, making it essential in simplifying expressions and finding solutions.

Understanding Integers

Integers include all whole numbers and their negative counterparts. They possess no fractional or decimal parts and are crucial in many mathematical operations.
This family of numbers includes positive numbers like 1, 2, and 3, zero, and negative numbers like -1, -2, and -3.

In our exercise, integers are used inside the parentheses.

• The integer addition in \(2+6\) involves two positive integers, which easily add up to 8.
• The addition of a negative and a positive integer, \(-5+6\), affects how we handle subtraction, resulting in 1.

Understanding how to combine and manipulate integers helps solve various expressions and equations efficiently.