Question

Find the value of each expression. \([-2+(-3)]^{2}+(2+3)^{2}\)

Short answer

The value of the expression is 50.

Step-by-step solution

Simplify the Expression inside the Brackets

First, simplify the expression inside the brackets: \(-2 + (-3)\) and \(2 + 3\). Calculate each part: \(-2 + (-3) = -5\) and \(2 + 3 = 5\). So, the expression becomes \((-5)^2 + (5)^2\).

Square the Results of Each Bracket

Now, square the results from Step 1:- For \((-5)^2\), calculate:\((-5)^2 = 25\).- For \((5)^2\), calculate:\((5)^2 = 25\).

Add the Squared Values Together

Add the squared values together:\(25 + 25 = 50\).

Key concepts

Exponents

Exponents represent a number multiplied by itself a certain number of times. This is indicated by a small number placed to the upper right of the base number. For example, in \((5)^2\), the base is 5 and the exponent is 2, meaning 5 is multiplied by itself once, resulting in 25.

Exponents have some key properties:

• **Power of One**: Any number to the power of one is itself, \(a^1 = a\).
• **Power of Zero**: Any non-zero number to the power of zero is 1, \(a^0 = 1\).
• **Negative Exponent**: This means reciprocal, \(a^{-b} = \frac{1}{a^b}\).

In the given expression, we encounter exponents while calculating \((-5)^2\) and \(5^2\). Here, 2 serves as the exponent, making it necessary to multiply the base number by itself once:

- \((-5)\times(-5) = 25\)

• Notice that multiplying two negative numbers results in a positive product.

- \((5)\times(5) = 25\)

• This operation yields a positive product as well.

Order of Operations

When solving mathematical expressions, it's critical to follow a systematic approach called the order of operations. This ensures consistent and correct solutions. This sequence is often remembered using the acronym PEMDAS:

• **P**arentheses
• **E**xponents (or powers)
• **M**ultiplication and **D**ivision (from left to right)
• **A**ddition and **S**ubtraction (from left to right)

In the expression \([-2+(-3)]^{2}+(2+3)^{2}\), the order of operations guides us to:1. Simplify within the brackets first: \(-2 + (-3) = -5\) and \(2 + 3 = 5\).2. Apply the exponents, resulting in \((-5)^2\) and \((5)^2\).3. Finally, proceed with the addition of squared values.Each step systematically simplifies the expression while adhering to the correct order, leading to the final result.

Integer Addition

Integer addition involves summing whole numbers, which can be positive, negative, or zero, to produce a resulting integer. This is a foundation of arithmetic and important when evaluating expressions or performing calculations such as within parentheses and before applying exponents.

Important considerations when adding integers:

• Adding a Positive Integer: Move right on a number line for each unit added. Example: \(5 + 3 = 8\).
• Adding a Negative Integer: Move left on a number line for each unit added. Example: \(5 + (-3) = 2\).
• Adding Two Negative Integers: The result takes the negative sign. Example: \(-4 + (-6) = -10\).

Returning to our expression \([-2+(-3)]^{2}+(2+3)^{2}\), integer addition is first used inside the brackets:

• \(-2 + (-3)\) becomes \(-5\); both integers are negative, resulting in a larger absolute value.
• \(2 + 3\) becomes \(5\); both integers are positive, resulting in a straightforward addition.

Understanding how integer addition works provides a solid foundation for solving complex expressions with ease.