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Chapter 2: Problem 94

Simplify the following problems. $$ 4^{2}+8 $$

Short Answer

Expert verified
Question: Simplify the expression \(4^2 + 8\). Answer: 24

Step by step solution

01

Evaluate the exponent

In the expression, we have the exponent \(4^2\). To calculate this, we have to multiply 4 by itself, 2 times. So, \(4^2 = 4\times 4 = 16.\)
02

Perform the addition

Now we can perform the addition in the expression. Replace \(4^2\) with the calculated value from step 1, resulting in \(16 + 8\).
03

Simplify the expression

Add the two numbers together to simplify the expression: \(16 + 8 = 24\). The simplified expression is \(24\).

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Key Concepts

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

Exponents
Exponents, also known as powers, are a mathematical way to express repeated multiplication of the same number. If you see an exponent such as \(4^2\), this is telling you to multiply the base number, 4, by itself the number of times indicated by the exponent, which is 2 in this case. Here, the calculation is \(4 \times 4 = 16\). Exponents make it easier to write complex multiplication briefly and understandably.

When simplifying expressions with exponents, remember:
  • The base is the number that gets multiplied.
  • The exponent tells you how many times to multiply the base by itself.
  • Calculating the exponent has the highest priority in mathematical operations, even before addition or subtraction.
Understanding how to handle exponents is crucial as they appear in a variety of mathematical problems, from algebra to sciences.
Arithmetic Operations
Arithmetic operations are fundamental mathematical processes that include addition, subtraction, multiplication, and division. Each of these operations has specific rules which govern how calculations are executed. In the problem given, addition is the arithmetic operation we need to perform after evaluating the exponent.

To solve the expression \(16 + 8\), we perform simple addition. Start by taking the result of the exponent calculation (16, as previously computed) and then add it to 8. It's a straightforward operation:
  • Align the numbers if necessary (especially in complex calculations).
  • Combine the numbers to get the result: \(16 + 8= 24\).
Performing arithmetic operations correctly is essential as these skills form the foundation of more complex mathematical concepts and applications.
Step-by-Step Solutions
Step-by-step solutions are a powerful way to understand and solve mathematical problems by breaking them down into manageable parts. This method offers clarity, especially in problems involving multiple operations such as the one in the exercise.

Let's break down why step-by-step solutions are helpful:
  • They guide you through the problem-solving process sequentially.
  • Allow you to focus on one operation at a time, reducing the chance of mistakes.
  • Provide a roadmap that you can follow or look back to if needed.
In our specific problem:1. Evaluating the exponent \(4^2\) gave us 16.2. Adding 16 to 8 simplified the expression to 24.

With step-by-step solutions, the complexity of a problem is reduced, making it easier to tackle even the toughest challenges in mathematics.

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Most popular questions from this chapter

Simplify the following problems. $$ \left(6 x^{4} y^{10}\right)\left(x y^{3}\right) $$

Use the product rule and quotient rule of exponents to simplify the following problems. Assume that all bases are nonzero and that all exponents are whole numbers. $$ \frac{x^{0}}{x^{0}} $$

Use the product rule and quotient rule of exponents to simplify the following problems. Assume that all bases are nonzero and that all exponents are whole numbers. $$ t^{2}\left(y^{4}\right) $$

Use the product rule and quotient rule of exponents to simplify the following problems. Assume that all bases are nonzero and that all exponents are whole numbers. $$ \frac{24 x^{4} y^{4} z^{0} w^{8}}{9 x y w^{7}} $$

For the following problems, write the expressions using exponential notation. $$ (x-9)(x-9)+(3 x+1)(3 x+1)(3 x+1) $$
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