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Chapter 1: Problem 78

Simplify the given expression. $$ 5^{3}-(17+15) $$

Short Answer

Expert verified
The simplified expression is 93.

Step by step solution

01

Calculate the Power

First, find the value of \(5^3\). To do this, multiply 5 by itself three times: \(5 \times 5 \times 5 = 125\).
02

Solve the Parentheses

Next, solve the expression inside the parentheses, \((17 + 15)\). Add the numbers together: \(17 + 15 = 32\).
03

Subtract the Results

Lastly, subtract the sum from the power: \(125 - 32\). Perform the subtraction to get \(125 - 32 = 93\).

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Key Concepts

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

Understanding Exponents
Exponents are a fundamental concept in mathematics that allow us to express repeated multiplication of a number by itself. For example, in the expression \(5^3\), 5 is the base, and 3 is the exponent. This means that we multiply 5 by itself three times: \(5 \times 5 \times 5\). The result of this multiplication is 125. This shorthand notation can simplify our work, especially with larger numbers.
  • An exponent indicates how many times to use a number in a multiplication.
  • \(5^3\) reads as "five raised to the power of three," meaning five is used as a factor three times.
Remember that understanding how to handle exponents is key, as they often appear in equations within algebra and other advanced math topics. Knowing how to simplify them can make solving complex expressions much easier.
Mastering the Order of Operations
The order of operations is a set of rules that tell us which arithmetic operations to perform first in a mathematical expression. To simplify complex expressions correctly, it's essential to follow these rules. The acronym PEMDAS is a useful tool to remember the order: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
  • First, complete the operations inside Parentheses.
  • Next, evaluate any Exponents.
  • Then, perform Multiplication and Division as they appear from left to right.
  • Finally, execute Addition and Subtraction from left to right.
In our exercise, solving \(5^3 - (17 + 15)\) begins with evaluating the exponent, followed by the operation inside the parentheses, and finally performing subtraction. Sticking to this order ensures you arrive at the correct solution.
Demystifying Arithmetic Expressions
Arithmetic expressions are combinations of numbers, operators, and sometimes parentheses that represent a particular value or operation. To simplify arithmetic expressions, it's crucial to carry out operations in the correct sequence, often guided by the order of operations.
  • Expressions can include operations such as addition, subtraction, multiplication, and division.
  • They often involve simplifying parts of the expression step by step.
  • Using the right order helps prevent errors in solving problems.
In the given example, the arithmetic expression \(5^3 - (17 + 15)\) breaks down into manageable steps, simplifying first the power and then dealing with the addition within the parentheses, before finishing with subtraction. Each step is critical to maintaining clarity and accuracy in getting to the final answer.

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