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Chapter 3: Problem 2

Write each of the following using only positive exponents. $$ m^{-2} $$

Short Answer

Expert verified
Question: Rewrite the expression \(m^{-2}\) with a positive exponent. Answer: \(\frac{1}{m^2}\)

Step by step solution

01

Identify the base and exponent

The given expression is \(m^{-2}\), where the base is \(m\) and the exponent is \(-2\).
02

Apply the rule of exponents

Using the rule of exponents mentioned in the analysis, we will rewrite the expression with a positive exponent: $$m^{-2} = \frac{1}{m^2}$$
03

Final answer

The expression \(m^{-2}\), rewritten with a positive exponent, is: $$\frac{1}{m^2}$$.

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Key Concepts

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

Negative Exponents
Negative exponents might seem tricky at first, but they follow a simple rule. When a number or variable is raised to a negative exponent, you can rewrite it as the reciprocal of the base raised to the corresponding positive exponent. For example, for an expression like \( m^{-2} \), you take the base, \( m \), and flip it to its reciprocal, raising it to the positive version of the exponent: \( \frac{1}{m^2} \). This means the base is in the denominator instead of the numerator.
  • The negative sign in the exponent indicates the reciprocal.
  • Only the position changes from numerator to denominator, not the value of the base.
  • This principle applies equally to numbers and variables.
As a tip, always remember that a negative exponent doesn't make the base negative; it only moves it to the other side of the fraction.
Positive Exponents
Positive exponents are generally the most straightforward to handle. They tell you how many times to multiply the base by itself. For instance, \( m^2 \) means \( m \times m \). Opposite to negative exponents, positive exponents keep the base in the numerator or at the same manipulative level you are working with.
  • Positive exponents indicate repeated multiplication of the base.
  • With \( m^2 \), the base \( m \) is multiplied by itself.
  • Exponents apply to the entire expression inside the parentheses.
This simplicity of positive exponents makes them a less daunting concept than their negative counterparts, offering clarity in expressions and equations.
Algebra
Algebra is a broad area of mathematics. It uses symbols, usually letters, to represent variables and constants. These symbols are then manipulated through various operations, including addition, subtraction, multiplication, division, and exponents. In expressions such as \( m^{-2} \), algebra helps simplify and solve for these symbols.
  • Algebra is the language of symbols and equations.
  • It allows for generalized operations instead of specific numeric calculations.
  • Through algebra, negative exponents can be rewritten in simplified forms.
A solid grasp of algebraic principles helps in reinterpreting equations, such as moving from a negative to a positive exponent.
Base and Exponent
A crucial concept in dealing with powers is understanding base and exponent. The base is the number or variable being multiplied, while the exponent tells you how many times to multiply the base by itself. For example, in \( m^{-2} \), \( m \) is the base, and \(-2\) is the exponent.
  • The base is the foundation for repeated multiplication.
  • The exponent signals how many times the base appears in the product.
  • Understanding base and exponent dimensions facilitates easier manipulation of expressions.
This concept helps in transforming expressions, ensuring they are correctly positionated, whether that involves flipping, multiplying, or dividing.

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Most popular questions from this chapter

For the following problems, convert the numbers from scientific notation to standard decimal form. Rubik's cube has about \(4.3 \times 10^{19}\) different configurations.

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