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Chapter 0: Problem 97

Write each algebraic expression without parentheses. $$-(-14 x)$$

Short Answer

Expert verified
The expression \(-(-14x)\) simplifies to \(14x\).

Step by step solution

01

Identifying the Expression

We have an expression \(-(-14x)\), which is a negation of a negative number.
02

Simplify the Expression Using the Rule of Signs

According to the rule of signs, when a negative sign is used on a negative number, it makes the value positive. Thus, the expression simplifies to \(14x\).

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Key Concepts

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

Negation
Negation is a fundamental concept in both mathematics and logic. It involves changing the sign of a number or reversing the truth value in logic. In algebra, negation specifically refers to changing a number's sign.
When a positive number is negated, it becomes negative, and vice-versa. For example:
  • Negation of 5 is (-5).
  • Negation of (-3) is 3.
In expressions, negation is depicted through the use of a minus sign (鈭�). If we see an expression like (-(-x)), it means we are applying negation twice. Each negation toggles the sign, so the pair of negatives effectively cancel each other out, resulting in a positive sign overall. This principle of negation is often useful in simplifying algebraic expressions.
Rule of Signs
The rule of signs in mathematics helps us determine the sign of a number based on the signs of the numbers being added, subtracted, multiplied, or divided.
In the scenario of multiplying or dividing two numbers:
  • Two positive numbers produce a positive result.
  • Two negative numbers also produce a positive result. This is pivotal for understanding how negation works with multiple negatives.
  • A positive and a negative number will produce a negative result.
For the expression (-(-14x)), applying the rule of signs helps us infer that since we are dealing with two negatives (the negative sign outside and the (-14x)), the outcome will be positive. Therefore, the expression simplifies to (14x). We can think of it like removing layers of negativity to reveal the underlying positive value.
Simplify Expression
Simplifying an expression means rewriting it in the most reduced form without changing its value. This often involves applying mathematical rules like distribution, combining like terms, or negation.
For instance, consider the expression (-(-14x)). Simplification steps:
  • Recognize the double negation.
  • Use the rule of signs: (-(-14x)) becomes (14x) because double negation turns it positive.
Simplification in algebra often makes expressions easier to understand and work with. The key is recognizing which rules to apply in different situations to reduce complexity without altering meaning.

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What is a linear equation in one variable? Give an example of this type of equation.

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a. A mathematics professor recently purchased a birthday cake for her son with the inscription $$ \text { Happy }\left(2^{\frac{5}{2}} \cdot 2^{\frac{3}{4}} \div 2^{\frac{1}{4}}\right) \text { th Birthday. } $$ How old is the son?b. The birthday boy, excited by the inscription on the cake, tried to wolf down the whole thing. Professor Mom, concerned about the possible metamorphosis of her son into a blimp, exclaimed, "Hold on! It is your birthday, so why not take \(\frac{8^{-\frac{4}{3}}+2^{-2}}{16^{-\frac{3}{4}}+2^{-1}}\) of the cake? I'll eat half of what's left over." How much of the cake did the professor eat?
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