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Chapter 1: Problem 99

Simplify each expression. $$(2+4)^{2}$$

Short Answer

Expert verified
The simplified expression is 36.

Step by step solution

01

Simplify the expression within the bracket

We first calculate the expression within the brackets. This is (2+4), which simplifies to 6.
02

Raise to the power

Now we take this result and raise it to the power. This gives us \(6^{2}\).
03

Final calculation

Finally, \(6^{2} = 36\). Hence the solution to the original expression \((2+4)^{2}\) is 36.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exponents
Exponents are a fundamental part of algebra that indicate repeated multiplication of a number by itself. Consider the expression \(a^b\) where \(a\) is the base, and \(b\) is the exponent or power. This expression signifies that the base \(a\) is multiplied by itself \(b\) times. In our exercise, the number 6 raised to the power 2 is represented as \(6^2\). This tells us that 6 is multiplied by itself once, which can be expanded to \(6 \times 6\). Understanding exponents is crucial for simplifying expressions, as it allows you to condense a sequence of multiplications into a compact form.
Order of Operations
The order of operations is a set of rules that dictates the correct sequence to evaluate a mathematical expression. Often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), it ensures uniformity in solving expressions.
  • Parentheses: Solve expressions within parentheses first.
  • Exponents: Calculate all exponential terms next.
  • Multiplication and Division: Proceed from left to right.
  • Addition and Subtraction: Lastly, handle these from left to right.
By following this order, the expression \((2+4)^2\) is solved by first addressing the parentheses \((2+4)\), which results in 6, before evaluating the exponent \(6^2\). Correctly applying the order of operations avoids mistakes and ensures accurate outcomes.
Simplification
Simplification involves reducing a mathematical expression to its most basic form. It is about performing all possible arithmetic operations to eliminate parentheses, solve exponents, and simplify addition, subtraction, multiplication, and division.
In the given expression \((2+4)^2\), simplification begins by solving the operation inside the parentheses: \(2+4\), resulting in 6. Then, move to the exponentiation step where the 6 is squared to get 36. These steps reduce the original expression into its simplest form: 36.
Simplification makes complex problems more manageable and helps ensure accurate results by systematically performing necessary calculations.

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