Problem 43 - Intermediate Algebra

Question

Simplify each of the numerical expressions. $$ (3+4)^{2} $$

Step-by-Step Solution

Answer

The simplified expression is 49.

Step 1: Calculate Inside the Parentheses

Start by performing the addition inside the parentheses. The expression inside is \(3 + 4\), so calculate \(3 + 4 = 7\).

Step 2: Apply the Exponent

Now take the result from Step 1, which is \(7\), and apply the exponent. The expression becomes \(7^{2}\). To calculate \(7^{2}\), multiply 7 by itself: \(7 \times 7 = 49\).

Step 3: Conclusion

The simplified expression of \((3+4)^{2}\) is \(49\).

Key Concepts

Understanding Parentheses in Expressions

Parentheses play a crucial role in determining the order in which operations are performed in mathematical expressions. They are used to group parts of an expression and denote that the operations inside them should be carried out first. This is because operations inside parentheses have a higher priority than those outside. For instance, in the expression \((3 + 4)^{2}\), you treat the values within the parentheses as a single unified number.

When simplifying an expression, always handle the operations within parentheses before moving on to others. Here are a few points about parentheses:

• Simplify everything inside the parentheses before dealing with exponents, multiplications, or additions outside of them.
• If there are nested parentheses, start with the innermost pair.
• Think of parentheses as a way to "set aside" certain operations until they are fully evaluated.

It’s almost like opening a present! You have to go through the layers in the right order to reach the gift inside.

Exploring Exponents

Exponents represent repeated multiplication of the same number. They are a shortcut to simplifying expressions where a number is multiplied by itself a certain number of times. In our example, when we write \( (3 + 4)^{2} \), the exponent \(2\) tells us to multiply the number inside the parentheses by itself.

Applying an exponent is straightforward once you have evaluated any parentheses. Here is what you need to remember about using exponents:

• The base - the number being multiplied - is taken from the result of evaluating inside the parentheses.
• The exponent tells us how many times to multiply the base by itself.
• If the exponent is \(2\), this is also known as "squared"; if \(3\), it is "cubed."

In our example, once you simplify the parentheses to \(7\), turning to \(7^{2}\) means calculating \(7 \times 7\), which equals \(49\). Exponents compactly convey powerful math.

Mastering the Order of Operations

The order of operations is a set of rules that dictate the sequence in which calculations should be done to ensure consistent results. A common acronym to remember this is PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Let's break down the order of operations:

• Parentheses: Address these first to simplify expressions within them.
• Exponents: Handle next, applying them to the numbers or results from parentheses.
• Multiplication/Division: Proceed to these, working from left to right, as they are evenly prioritized.
• Addition/Subtraction: Finally, complete these remaining operations, also working from left to right.

This clear guideline resolves potential ambiguities in mathematical calculations. In our problem, we start with the parentheses \((3+4)\), evaluate to \(7\), then apply the exponent to get \(49\). Remember: Follow PEMDAS, and you’ll always reach the correct result!