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Chapter 1: Problem 79

Simplify each expression. $$(-1)\left(2^{3}\right)$$

Short Answer

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-8

Step by step solution

01

Evaluate the Exponential Term

The expression contains an exponential term \(2^3\). According to the rule of exponents, \(2^3\) means multiplying 2 by itself 3 times, which gives: \(2^3 = 2 脳 2 脳 2 = 8\). So the expression becomes:\n\(-1\) (8)
02

Compute the Multiplication

With the exponential term evaluated, now it's time to do the multiplication. Multiply -1 and 8 which gives: \(-1 脳 8 = -8\).

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Key Concepts

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

Exponents
Understanding exponents is crucial in simplifying expressions. An exponent tells us how many times we use the base number as a factor in multiplication. In the expression \(2^3\), the base is 2 and the exponent is 3, which means you multiply 2 by itself three times: \(2 \times 2 \times 2\). Each multiplication step can be broken down as follows:
  • First, multiply 2 by 2 to get 4.
  • Then, take the result, 4, and multiply it by 2 again to get 8.
So, \(2^3 = 8\). This process is known as exponentiation, where the base is 2 and the number of times it's multiplied by itself is denoted by the exponent. Exponents are a shorthand way of expressing repeated multiplication, making calculations easier and more elegant.
Multiplication
After evaluating the exponential part of the expression, the next step involves multiplication. In mathematical operations, multiplication is essentially repeated addition. In our example, we have the number 8 from the evaluated exponent, which is to be multiplied by -1.
  • When multiplying two numbers, one of them being negative, the result will always be negative.
  • The rule can be simplified as: a positive number multiplied by a negative number results in a negative product.
So, when you multiply \(-1 \times 8\), you arrange it as the addition of -8 among 1 unit of positive 8: it results in \(-8\). Multiplying by negative one essentially "flips" the sign of the positive number, converting positive 8 into negative 8.
Negative Numbers
Negative numbers play an interesting role in mathematics, especially noticeable in operations like multiplication. A negative number is simply a number that is less than zero, represented with a minus sign. Here are some points to help understand negativity with ease:
  • Negative numbers are the opposite of positive numbers and appear to the left of zero on a number line.
  • Multiplying a positive number by a negative number gives a negative result.
  • Conversely, when you multiply two negative numbers, the outcome is positive because two negatives make a positive.
In our expression, the factor -1 implies that a positive number like 8 will be changed to its negative equivalent after multiplication. Therefore, \(-1 \times 8\) changes the positivity of 8 to -8, mirroring it across zero on the number line and giving us the final simplified negative result.

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