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Chapter 0: Problem 38

Evaluate each expression. $$ -10^{3} $$

Short Answer

Expert verified
-1000

Step by step solution

01

Identify the Base

In the expression \(-10^{3}\), the base is \(-10\). This is the number that will be raised to the power of 3.
02

Identify the Exponent

The exponent in the expression is \(3\). This means the base \(-10\) must be multiplied by itself three times.
03

Calculate the Power

Raise \(-10\) to the power of \(3\) by multiplying: \(-10 \times -10 \times -10 = -1000\).
04

Write the Final Answer

The value of the expression \(-10^{3}\) is \(-1000\).

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Key Concepts

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

Base and Exponent
In any mathematical expression involving exponents, two main components come into play: the base and the exponent. The base is the number that is going to be multiplied, and the exponent tells you how many times the base will be used in the multiplication.
For example, in the expression \(2^5\), 2 is the base and 5 is the exponent. This tells us that 2 should be multiplied by itself 5 times, which can be written as \(2 \times 2 \times 2 \times 2 \times 2\).
Understanding the role of both base and exponent is crucial for evaluating expressions correctly.
Power Calculation
Power calculation involves raising a base to the power of an exponent. Let's revisit the original exercise: \(-10^3\). Here, -10 is the base and 3 is the exponent.
To calculate the power, you need to multiply the base by itself as many times as indicated by the exponent. In our example, we multiply \(-10 \times -10 \times -10\). Calculating step-by-step, we find:
\(-10 \times -10 = 100\)
\(100 \times -10 = -1000\)
Therefore, \(-10^3 = -1000\).
So, always follow these steps:
  • Identify the base and exponent.
  • Multiply the base by itself for the number of times given by the exponent.
  • Check the sign (positive or negative) of your result based on the number of negatives involved.
Negative Numbers
Negative numbers are numbers with a value less than zero, and they can change the outcome of multiplication, depending on how many times they are used.
Multiplying two negative numbers results in a positive number: \( -2 \times -3 = 6 \).
However, when you multiply a negative number by a positive number, the result is negative: \( -2 \times 3 = -6 \).
For exponentiation, when a negative base is raised to an odd power, the result is negative (as in \(-10 \times -10 \times -10 = -1000\)). But for an even power, the result is positive:
  • \( (-2)^3 = -8 \)
  • \( (-2)^4 = 16 \)
Knowing these properties helps to avoid common mistakes when dealing with exponents and negative numbers.

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