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Chapter 1: Problem 59

Simplify each expression. See Examples I through 11 . $$ (-3)^{2}+2^{3} $$

Short Answer

Expert verified
The expression simplifies to 17.

Step by step solution

01

Simplify the Exponents

Calculate each exponent separately. First, for \((-3)^2\), square the number -3:\((-3) \times (-3) = 9\).Next, calculate the cube of 2, which is \(2^3 = 2 \times 2 \times 2 = 8\).
02

Add the Results

Now that we have simplified the exponents to 9 and 8, add them together:\(9 + 8 = 17\).

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Key Concepts

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

Exponents
Exponents are a convenient way to express repeated multiplication of the same number. For example, if we have the expression \((-3)^2\), the exponent "2" indicates we multiply the base number "-3" by itself. That means \((-3) \times (-3) = 9\). Similarly, an exponent of "3" suggests the number is multiplied by itself three times. For instance, \(2^3\) is equal to \(2 \times 2 \times 2 = 8\).Here鈥檚 why understanding exponents is important:
  • They simplify expressions involving repeated multiplication.
  • Exponents affect the overall value of expressions considerably, depending on the base and power.
Whenever you simplify expressions with exponents, always handle the exponents first before moving onto other operations. This approach helps keep calculations organized and accurate.
Adding
Once you've simplified the parts of an expression with exponents, the next step is often to add the simplified values.In our example, we calculated the exponents to reach values of 9 and 8. Now, we add these two numbers together: \(9 + 8 = 17\).Adding is one of the basic arithmetic operations that combines two or more numbers to get a sum. When simplifying expressions, it's crucial to
  • Ensure all parts to be added are fully simplified, especially if they involve other operations like exponents or multiplication.
  • Double-check each step to avoid small mistakes that can affect the final answer.
Adding correctly is vital in ensuring that the result of simplifying any mathematical expression is accurate.
Order of Operations
Understanding the order of operations is essential when dealing with complex mathematical expressions. The order dictates which operations must be performed first to obtain the correct result. A common way to remember this order is PEMDAS:
  • **P**arentheses - complete expressions inside parentheses first.
  • **E**xponents - resolve exponents next.
  • **M**ultiplication and **D**ivision - perform from left to right.
  • **A**ddition and **S**ubtraction - also from left to right.
In the given problem, we first resolved the exponents:\((-3)^2 = 9\) and \(2^3 = 8\).Following this, we performed addition:\(9 + 8\).By adhering to the order of operations, you ensure that every expression you simplify will lead to the correct answer, minimizing errors in calculations.

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