Power

Definition

In mathematics, power refers to the expression of a number raised to an exponent, which indicates how many times the base number is multiplied by itself. This concept allows for the simplification of large numbers and plays a vital role in operations involving exponential growth, roots, and logarithms. Understanding power helps in grasping how numbers can be manipulated in various mathematical contexts.

5 Must Know Facts For Your Next Test

1. The notation for power is typically written as $$a^n$$, where 'a' is the base and 'n' is the exponent.
2. A power with an exponent of 1 simply equals the base itself (e.g., $$a^1 = a$$).
3. Any non-zero number raised to the power of 0 equals 1 (e.g., $$a^0 = 1$$).
4. Negative exponents indicate reciprocal values; for instance, $$a^{-n} = \frac{1}{a^n}$$.
5. Powers can be multiplied and divided using rules such as $$a^m \cdot a^n = a^{m+n}$$ and $$\frac{a^m}{a^n} = a^{m-n}$$.

Review Questions

• How do you apply the properties of powers when multiplying two numbers with the same base?
  • When multiplying two numbers with the same base, you can apply the property that states $$a^m \cdot a^n = a^{m+n}$$. This means you simply add the exponents together while keeping the base unchanged. For example, if you multiply $$2^3$$ and $$2^4$$, you would calculate it as $$2^{3+4} = 2^7$$.
• What is the significance of negative exponents in terms of their relationship with positive exponents?
  • Negative exponents are significant because they provide a way to express division by a number raised to a positive exponent. For example, if you have a negative exponent like $$a^{-n}$$, it is equivalent to $$\frac{1}{a^n}$$. This concept is useful in simplifying fractions and understanding reciprocal relationships in mathematics.
• Evaluate how understanding powers can enhance problem-solving abilities in real-world applications such as finance or science.
  • Understanding powers significantly enhances problem-solving abilities in real-world applications by allowing individuals to work with exponential growth and decay effectively. For example, in finance, compound interest calculations utilize powers to determine future values based on present investments. Similarly, in science, exponential functions describe phenomena like population growth or radioactive decay. Mastery of powers enables clearer insights into these complex situations, improving decision-making and predictions.

Related terms

Exponent: An exponent is a mathematical notation that indicates how many times a number, known as the base, is multiplied by itself.

Base: The base is the number that is being raised to a power or exponent in an expression.

Square:

A square refers to the power of 2, meaning a number multiplied by itself, often represented as 'n^2' where 'n' is the base.