Exponents with fractional bases

Learning Objectives Identify the base and exponent in expressions with fractional or decimal bases. Explain the meaning of an exponent when the base is a fraction or a decimal. Calculate the value of expressions with fractional bases raised to positive integer exponents. Convert fractional bases to decimal form and vice versa to simplify calculations. Apply the order of operations when solving problems involving exponents with fractional bases. Solve real-world problems that require calculating powers of fractional or decimal bases. Ever wondered how scientists calculate the growth of bacteria that doubles every hour, or how architects scale down building plans? 🤔 It often involves multiplying fractions or decimals by themselves! In this lesson, you'll learn how to wo...

TermDefinitionExample BaseThe number that is being multiplied by itself in an exponential expression.In $(1/2)^3$, $1/2$ is the base. ExponentThe small number written above and to the right of the base, indicating how many times the base is to be multiplied by itself.In $(0.75)^2$, $2$ is the exponent. PowerThe entire expression consisting of a base and an exponent, or the result of evaluating such an expression.$(1/2)^3$ is a power, and its value, $1/8$, is also a power. Fractional BaseA base that is written as a common fraction (e.g., $rac{3}{4}$) or a mixed number.$(\frac{2}{3})^4$ has a fractional base. Decimal BaseA base that is written as a decimal number (e.g., $0.5$).$(0.2)^3$ has a decimal base. ProductThe result obtained when two or more numbers are multiplied together.The prod...

Definition of Exponentiation $a^n = a \times a \times ... \times a$ (n times) This rule defines what an exponent means. The base 'a' is multiplied by itself 'n' times. This applies whether 'a' is a whole number, a fraction, or a decimal. Power of a Fraction $(\frac{a}{b})^n = \frac{a^n}{b^n}$ When a fraction is raised to an exponent, both the numerator and the denominator are raised to that exponent. This can simplify calculations. Converting Fraction to Decimal $\frac{a}{b} = a \div b$ To convert a fractional base to a decimal base, divide the numerator by the denominator. This is often useful for calculations, especially with calculators.