4 Questions
If a = 5^2 and c = 5^3, what is ac equivalent to?
5^5
What is the result of (3^4) * (3^2)?
3^6
What does (6^4) / (6^2) simplify to?
6^2
If x = 2^3 and y = 2^5, what does x/y simplify to?
2^-2
Study Notes
Exploring Maths Chapter: Exponents and Power with Laws of Exponents
In the realm of mathematics, "exponents and power" are essential topics that introduce the concept of raising a number or expression to a power or raising it to a certain degree. This article delves into the laws of exponents, focusing on the subtopics within this subject area.
Exponents Exponents are a compact way of writing repeated multiplication. For instance, instead of writing 2 x 2 x 2 x 2 x 2, we can represent this as 2^5 (read as "two to the power of five"). The exponent is the number (called the "base") being multiplied by itself.
Laws of Exponents The laws of exponents are a set of rules that govern how to simplify expressions involving exponents. These rules apply to any real number base, whether integers, fractions, or even radicals.
Here are some key laws of exponents:
- Product of Powers: If a = b^m and c = b^n, then ac = b^(m+n)
- Quotient of Powers: If a = b^m and c = b^n, then a/c = b^(m-n)
- Power of a Power: If a = b^m, then a^n = b^(mn)
- Negative Exponents: If a ≠ 0, then a^(-n) = 1/(a^n)
- Fractional Exponents: a^(1/n) is the nth root of a
- Zero Exponent: a^0 = 1 (for any non-zero a)
- Exponent of a Product: (ab)^n = a^n * b^n
- Exponent of a Quotient: (a/b)^n = a^n / b^n
- Exponent of a Power: (a^m)^n = a^(mn)
Examples
- 3^2 * 3^3 = 3^(2+3) = 3^5 = 243
- (4^3) / (4^2) = 4^(3-2) = 4^1 = 4
- (2^4) * (2^3) = 2^(4+3) = 2^7 = 128
- -2^3 = -2 * 2 * 2 = -8
- √81 = 3 (since 3 * 3 = 9, which is 81's square root)
- (2^3) * (3^2) = 8 * 9 = 72
- (5x)^2 = 5^2 * x^2 = 25x^2
These laws of exponents serve as fundamental tools for simplifying and solving mathematical problems. They're essential for understanding expressions with exponents, which are commonly used in algebra, geometry, and calculus.
Remember, these laws apply to any real number base, and they make calculations involving exponents more efficient and easier to understand. Mastering these rules will provide a solid foundation for further mathematical exploration.
Explore the realm of exponents and power in mathematics through the laws of exponents. Learn about the rules that govern simplifying expressions involving exponents and apply them to various examples and scenarios. Mastering these fundamental tools is crucial for solving mathematical problems in algebra, geometry, and calculus.
Make Your Own Quizzes and Flashcards
Convert your notes into interactive study material.
Get started for free