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Chapter 2: Problem 54

Simplify. See Examples 1 through \(10\). $$ -3(4-8)^{2}+5(14-16)^{3} $$

Short Answer

Expert verified
The simplified expression is \(-88\).

Step by step solution

01

Simplify Inside Parentheses

First, simplify the expressions inside each set of parentheses. For \(4 - 8\), subtract to get \(-4\); and for \(14 - 16\), subtract to get \(-2\). The expression becomes: \[-3(-4)^{2} + 5(-2)^{3}.\]
02

Apply Exponents

Next, apply the exponents to each of the simplified numbers. Calculate \((-4)^{2}\) which is \((-4) \times (-4) = 16\), and \((-2)^{3}\) which is \((-2) \times (-2) \times (-2) = -8\). The expression becomes: \[-3(16) + 5(-8).\]
03

Multiply Coefficients

Now, multiply the coefficients outside the parentheses by their respective results. Calculate \(-3 \times 16\) which equals \(-48\), and \(5 \times (-8)\) which equals \(-40\). The expression becomes:\[-48 - 40.\]
04

Combine Terms

Lastly, combine the terms \(-48\) and \(-40\) by adding them. \(-48 + (-40) = -88.\) The final simplified result is \(-88\).

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Key Concepts

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

Order of Operations
Understanding the order of operations is crucial in simplifying expressions accurately. It is often abbreviated as PEMDAS, standing for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
  • Start by simplifying expressions within parentheses.
  • Next, calculate any exponents.
  • Then, perform any multiplication or division as they appear from left to right.
  • Finally, carry out any addition or subtraction from left to right.
Following this sequence ensures that mathematical expressions are simplified the same way every time. The importance of this order cannot be overstated, as changing it can drastically alter the results of an expression.
Parentheses in Algebra
Parentheses play a vital role in algebra by indicating which operations should be performed first. In an expression, what鈥檚 inside the parentheses gets priority over other operations.

- For instance, in the problem \(-3(4-8)^{2}+5(14-16)^{3}\), \(4-8=-4\) and \(14-16=-2\) should both be simplified first, clarifying the next steps.
  • This ensures that calculations are done in the correct order.
  • Neglecting parentheses can lead to mistakes in calculations.
In practice, once the parentheses are simplified, you can proceed with other operations, such as using exponents and multiplying coefficients.
Exponents
Exponents are a way of indicating that a number, known as the base, should be multiplied by itself a certain number of times. For instance, \((-4)^{2}\) means \((-4) imes (-4)\).

- Exponents are always calculated immediately after any necessary parentheses are simplified.
In the provided problem:
  • Calculate \((-4)^{2}\) to get 16.
  • Calculate \((-2)^{3}\) to get -8.
Applying exponents accurately is critical, as they have a large influence on the final numerical value of any expression. Once exponents are calculated, you proceed with other operations like multiplication and addition, following the order of operations.

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