Exponents are mathematical notations that indicate the number of times a base number is multiplied by itself. They are fundamental in simplifying calculations and expressing large numbers concisely, playing a crucial role in various mathematical operations, including algebra and trigonometry. Understanding exponents allows for easier manipulation of equations and functions, especially when dealing with powers and roots.
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Exponents can be positive, negative, or zero, each having distinct meanings; for example, any non-zero number raised to the power of zero equals one.
When multiplying numbers with the same base, you can add their exponents; for instance, $$a^m imes a^n = a^{m+n}$$.
When dividing numbers with the same base, you subtract the exponents; thus, $$a^m ÷ a^n = a^{m-n}$$.
Exponent rules also include the power of a power rule, where $$(a^m)^n = a^{mn}$$.
Exponents are especially useful in scientific notation to express very large or very small numbers efficiently.
Review Questions
How do you apply the rules of exponents when multiplying two numbers with the same base?
When multiplying two numbers that share the same base, you apply the rule that states you should add their exponents together. For example, if you have $$2^3$$ and $$2^4$$, you would multiply them as follows: $$2^3 imes 2^4 = 2^{3+4} = 2^7$$. This property simplifies calculations significantly when working with exponential expressions.
What happens when you raise a number with an exponent to another exponent? Give an example.
When raising a number with an exponent to another exponent, you multiply the exponents together according to the power of a power rule. For instance, if you have $$(3^2)^3$$, you would compute it as follows: $$3^{2 imes 3} = 3^6$$. This simplifies the expression and makes it easier to work with larger powers.
Evaluate the expression $$5^{-2}$$ and explain what this indicates about negative exponents.
$$5^{-2}$$ evaluates to $$\frac{1}{5^2}$$ or $$\frac{1}{25}$$. This shows that negative exponents indicate the reciprocal of the base raised to the corresponding positive exponent. Understanding this concept is crucial because it helps in simplifying expressions involving negative exponents and clarifies how they operate mathematically.
Related terms
Base: The base is the number that is being multiplied in an exponent expression. For example, in $$2^3$$, 2 is the base.
Power: Power refers to the result of raising a base to an exponent. In the case of $$2^3$$, the power is 8.
Square Root: The square root is a value that, when multiplied by itself, gives the original number. It is often represented using fractional exponents, such as $$x^{1/2}$$.