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Chapter 1: Problem 63

Perform each operation. SEE EXAMPLE \(5 .\) (OBJECTIVE 2 ) $$\frac{(-2)^{3}(10)}{-(-5)}$$

Short Answer

Expert verified
-16

Step by step solution

01

Compute Exponent

The first step is to compute the exponent. So, \( (-2)^3 = -2*-2*-2 = -8 \)
02

Multiply Result by 10

The next step is to multiply the result by 10. So, \( -8 * 10 = -80 \)
03

Division Computation

Finally, the result of the multiplication is divided by -(-5). Changing the double negative to positive gives us 5. So, \( -80 / 5 = -16 \)

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Key Concepts

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

Exponentiation
Exponentiation is an important algebraic operation. It involves raising a number to the power of another number. In this context, the base number is multiplied by itself as many times as the exponent indicates. For instance, in the expression
  • \((-2)^3\), the base is -2 and the exponent is 3. This means that -2 is multiplied by itself three times:

  • (-2) \times (-2) \times (-2).
Negative bases require careful attention. When you raise a negative number to an odd power, the result is negative. For our problem,
  • \((-2)^3 = -8\). As -2 is multiplied three times, the result reflects the sign of the base number.
Understanding the difference between raising a number with a positive base versus a negative base to a power is crucial for solving algebraic expressions accurately.
Multiplication
Multiplication is a fundamental arithmetic operation that involves combining numbers in a process of repeated addition. In algebra, it takes on a broader sense where it adds structure to expressions and equations. If you take our previous result from exponentiation,
  • \(-8\), and multiply by 10, the expression is written as \(-8 \times 10\).
Performing this multiplication gives
  • \(-80\). The result adopts the sign of the operand that is not positive because multiplication of a negative and a positive number results in a negative product.
Through multiplication, single entities become compounded, and maintaining the correct signs ensures that algebraic solutions are logical and sound with their numerical foundations.
Division
Division is the arithmetic process used for finding out how many times one number is contained within another. In algebraic terms, it simplifies expressions by reducing complexity. Returning to our solution, we divide our multiplication result by a reformulated divisor.To handle the division \(-80 / -(-5)\), rewrite the divisor correctly. The double negative becomes a positive, transforming \(-(-5)\) to 5.
  • Divide \(-80 \times 5 = -16\).
      • This operation reveals that a negative divided by a positive remains negative. Each expression part holds its identity. Remember, division, unlike multiplication, handles the role of 'sharing' or distributing equally. Proper handling of signs in division ensures that operations align with the natural order of mathematics.

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