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Chapter 5: Problem 23

Simplify each expression. Write each result using positive exponents only. $$ -2^{0}-3^{0} $$

Short Answer

Expert verified
The expression simplifies to -2.

Step by step solution

01

Apply the Zero Exponent Rule

According to the Zero Exponent Rule, any non-zero number raised to the power of zero is equal to one. Thus, we simplify each term separately: \(-2^0 = -1\) and \(-3^0 = -1\).
02

Add the Results

Now, add the results obtained from Step 1: \(-1 - 1\). This computation simplifies to \(-2\).

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Key Concepts

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

Exponentiation
Exponentiation is a mathematical operation involving two numbers, the base and the exponent. The base is the number that is multiplied by itself, and the exponent shows how many times it is multiplied. Exponentiation is represented as \(a^n\). Here, \(a\) is the base, and \(n\) is the exponent.
  • For example, \(2^3\) means "2 multiplied by itself 3 times," which equals 8.
  • It simplifies the representation of large and repeated products.
  • This concept is essential for learning more advanced math topics.
Understanding the behavior of exponents is crucial. Some special rules exist, like the Zero Exponent Rule, which states any non-zero number raised to the power of zero equals one. This is because starting with the base multiplied by itself zero times leaves us with the multiplicative identity.
Simplifying Expressions
Simplifying expressions involves reducing a mathematical expression to its simplest form. This process often has us applying various mathematical rules to make calculations easier or more intuitive.
  • Simplification can include combining like terms, applying exponent rules, and reducing fractions.
  • The goal is to achieve the most compact and straightforward form without altering the expression's value.
When it involves exponents, knowing the rules simplifies the expression efficiently. The step-by-step approach is vital, ensuring each operation adheres strictly to arithmetic rules for consistency and correctness, like applying the Zero Exponent Rule.
Positive Exponents
Positive exponents are exponents that are greater than zero. These exponents tell us that we need to multiply the base by itself a certain number of times.
  • For instance, \(5^2\) means \(5\times5\), resulting in 25.
  • Positive exponents convey straightforward multiplication.
  • It's important to express results using positive exponents because they are the standard convention.
In problems where you need to rewrite expressions with positive exponents, it's helpful to first convert negative exponents using rules, then simplify accordingly. This helps clarify the expression, making it easier to understand and work with in future calculations.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operation symbols. They represent a mathematical phrase that can involve various operations like addition, subtraction, multiplication, and division.
  • An algebraic expression like \(3x + 5\) implies a relationship between numbers and variables.
  • Expressions often require simplification to make them more manageable.
  • Exponents are commonly part of algebraic expressions, influencing their simplification process.
In algebra, adhering to specific operations rules ensures a consistent and correct simplification. When simplifying, we may use fundamental rules, like zero powers, and combine like terms to streamline the expression comprehensively.

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Most popular questions from this chapter

Simplify each expression. Assume that variables represent positive integers. $$ a^{-4 m} \cdot a^{5 m} $$

Add \(\left(-3 x^{2}-5 x+2\right)\) and \(\left(x^{2}-6 x+9\right)\)

Subtract. $$ \left(\frac{1}{4} z^{2}-\frac{1}{5} z\right)-\left(-\frac{3}{20} z^{2}+\frac{1}{10} z-\frac{7}{20}\right) $$

Subtract \(\left(4 x^{2}-2 x+2\right)\) from the sum of \(\left(x^{2}+7 x+1\right)\) and \((7 x+5)\)

Simplify each expression by performing the indicated operation. Explain how you arrived at each answer. a. \(m \cdot m \cdot m\) b. \(m+m+m\) C. \((-m)(-m)(-m)\) d. \(-m-m-m\)
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