Exponents with decimal and fractional bases

Learning Objectives Define and identify the base and exponent when working with decimal and fractional numbers. Calculate the value of expressions with decimal bases raised to a positive integer exponent. Calculate the value of expressions with fractional bases raised to a positive integer exponent. Simplify expressions involving both decimal and fractional bases. Apply the concept of exponents with decimal and fractional bases to solve simple real-world problems. Recognize common pitfalls and apply correct strategies when evaluating powers of decimals and fractions. Ever wondered how scientists calculate the growth of tiny bacteria or how architects scale down huge buildings into small models? 🤔 It often involves special numbers called exponents! In this lesson, we'l...

TermDefinitionExample BaseThe number that is multiplied by itself in an exponential expression. In this lesson, the base can be a decimal or a fraction.In $(0.5)^3$, the base is $0.5$. In $(rac{2}{3})^4$, the base is $rac{2}{3}$. ExponentThe small number written above and to the right of the base, indicating how many times the base is used as a factor.In $(0.5)^3$, the exponent is $3$. In $(rac{2}{3})^4$, the exponent is $4$. PowerThe entire expression consisting of a base and an exponent, or the result of evaluating that expression.$(0.5)^3$ is a power. The value $0.125$ is also a power. Decimal BaseA base that is a decimal number (e.g., $0.2$, $1.5$, $3.75$).$(0.2)^2 = 0.2 imes 0.2 = 0.04$ Fractional BaseA base that is a fraction (e.g., $rac{1}{2}$, $rac{3}{4}$, $2rac{1}{3}$).$(...

Definition of Exponentiation $b^n = b \times b \times \dots \times b$ (n times) This fundamental rule applies to all bases, including decimals and fractions. The exponent 'n' tells you how many times to multiply the base 'b' by itself. Exponents with Fractional Bases $(\frac{a}{b})^n = \frac{a^n}{b^n}$ When a fraction is raised to an exponent, both the numerator and the denominator are raised to that exponent. This simplifies the multiplication process. Exponents with Decimal Bases To calculate $D^n$ where D is a decimal: Multiply D by itself 'n' times. The total number of decimal places in the product will be 'n' times the number of decimal places in D. This rule helps ensure accuracy when multiplying decimals. For example, if...