Laws of exponents

Definition

The laws of exponents are a set of rules that describe how to simplify expressions involving powers of numbers or variables. These laws help in performing operations like multiplication, division, and raising powers to powers, ensuring consistency and efficiency in mathematical calculations involving exponents.

5 Must Know Facts For Your Next Test

1. One key law states that when multiplying two powers with the same base, you can add their exponents: $$a^m \cdot a^n = a^{m+n}$$.
2. When dividing two powers with the same base, subtract the exponents: $$\frac{a^m}{a^n} = a^{m-n}$$.
3. Raising a power to another power involves multiplying the exponents: $$(a^m)^n = a^{m \cdot n}$$.
4. Any non-zero number raised to the power of zero equals one: $$a^0 = 1$$.
5. Negative exponents represent the reciprocal of the base raised to the positive exponent: $$a^{-n} = \frac{1}{a^n}$$.

Review Questions

• How do the laws of exponents apply when multiplying two expressions with the same base?
  • When multiplying two expressions with the same base, you apply the law that states you should add their exponents. For example, if you have $$2^3 \cdot 2^4$$, you would add the exponents to get $$2^{3+4} = 2^7$$. This makes it easy to simplify expressions quickly without having to calculate each power separately.
• Why is understanding negative exponents important in simplifying mathematical expressions?
  • Understanding negative exponents is crucial because they indicate the reciprocal of the base raised to a positive exponent. For example, $$a^{-3}$$ means $$\frac{1}{a^3}$$. This concept helps in simplifying complex fractions and allows for proper manipulation of expressions during calculations, especially in algebra and calculus.
• Evaluate and explain how applying laws of exponents can simplify complex algebraic expressions.
  • Applying laws of exponents can greatly simplify complex algebraic expressions by reducing them into more manageable forms. For instance, when simplifying an expression like $$\frac{x^5 \cdot x^{-2}}{x^3}$$, you can use multiplication and division laws to combine and simplify: first, multiply to get $$x^{5-2} = x^3$$, then divide by $$x^3$$ which results in 1. This shows how using these laws leads to a clearer understanding and easier computation in algebra.

Related terms

Exponent: An exponent indicates how many times a number, called the base, is multiplied by itself.

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

Scientific Notation: A way of expressing numbers that are too large or too small using powers of ten, which utilizes the laws of exponents for simplification.