Exploring the fundamentals of exponents in mathematics, this overview covers the laws of exponents, the relationship between roots and powers, and their practical applications. It delves into the product of powers rule, quotient of powers rule, and the power of a power rule. Understanding fractional exponents and their interplay with roots is essential for simplifying complex algebraic expressions, as is the ability to simplify expressions using the rules for powers and roots.
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Exponents are superscripts that indicate how many times a base number is used as a factor in repeated multiplication
Product of Powers Rule
When multiplying like bases, the exponents are added
Quotient of Powers Rule
When dividing like bases, the exponents are subtracted
Power of a Power Rule
When raising a power to another power, the exponents are multiplied
Negative exponents represent the reciprocal of the base raised to the positive exponent, while fractional exponents represent roots
Roots are the inverse operations of exponents and are used to find the original number when raised to a certain power
Square Root
The square root of a number has two solutions (positive and negative) for positive radicands
Cube Root
The cube root of a number has one real solution, even for negative radicands
Roots can be simplified using fractional exponents, with perfect square roots resulting in integers and non-perfect square roots resulting in irrational numbers
The rules for powers and roots can be applied to simplify algebraic expressions, such as recognizing common factors and using the quotient of powers rule
00
In math, if a number is raised to the power of 3, like x³, it signifies that x is ______ by itself ______ times.
multiplied
two
01
A number with an exponent of 4, written as x⁴, is x ______ by itself ______ times, while x to the zero power (x⁰) is always ______.
multiplied
three
1
02
Product of Powers Rule
Multiply like bases: add exponents (xᵃ · xᵇ = xᵃ⁺ᵇ).
