Learn on PengiBig Ideas Math, Course 2, AcceleratedChapter 6: Exponents and Scientific Notation

Lesson 1: Exponents

In this Grade 7 lesson from Big Ideas Math Course 2 Accelerated, students learn to identify and work with powers, bases, and exponents by writing repeated multiplication in exponential form and evaluating expressions with positive, negative, and fractional bases. The lesson also covers applying order of operations with exponents and using powers in real-life contexts such as calculating volumes of spheres.

Section 1

Exponents

Property

An exponent is a number that appears above and to the right of a particular factor. It tells us how many times that factor occurs in the expression. The factor to which the exponent applies is called the base, and the product is called a power of the base.
An exponent indicates repeated multiplication.

an=a⋅a⋅a⋯a(n factors of a)a^n = a \cdot a \cdot a \cdots a \quad (n \text{ factors of } a)

where nn is a positive integer.

Examples

  • To compute 535^3, we multiply three factors of 5: 5⋅5⋅5=1255 \cdot 5 \cdot 5 = 125.
  • The expression (14)2(\frac{1}{4})^2 means 14⋅14=116\frac{1}{4} \cdot \frac{1}{4} = \frac{1}{16}.

Section 2

Order of Operations with Exponents

Property

When evaluating expressions containing exponents, follow the order of operations (PEMDAS): Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Pay special attention to the difference with negative bases:

  • (−a)n(-a)^n means the base is −a-a
  • −an-a^n means the base is aa, and you take the opposite of the result

Section 3

Application: Evaluating Formulas with Exponents

Property

Many formulas in geometry and science use exponents. To evaluate these formulas, substitute the given values for the variables and then use the order of operations to simplify the expression. Common formulas include the area of a circle, A=πr2A = \pi r^2, and the volume of a cube, V=s3V = s^3.

Examples

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Exponents

Property

An exponent is a number that appears above and to the right of a particular factor. It tells us how many times that factor occurs in the expression. The factor to which the exponent applies is called the base, and the product is called a power of the base.
An exponent indicates repeated multiplication.

an=a⋅a⋅a⋯a(n factors of a)a^n = a \cdot a \cdot a \cdots a \quad (n \text{ factors of } a)

where nn is a positive integer.

Examples

  • To compute 535^3, we multiply three factors of 5: 5⋅5⋅5=1255 \cdot 5 \cdot 5 = 125.
  • The expression (14)2(\frac{1}{4})^2 means 14⋅14=116\frac{1}{4} \cdot \frac{1}{4} = \frac{1}{16}.

Section 2

Order of Operations with Exponents

Property

When evaluating expressions containing exponents, follow the order of operations (PEMDAS): Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Pay special attention to the difference with negative bases:

  • (−a)n(-a)^n means the base is −a-a
  • −an-a^n means the base is aa, and you take the opposite of the result

Section 3

Application: Evaluating Formulas with Exponents

Property

Many formulas in geometry and science use exponents. To evaluate these formulas, substitute the given values for the variables and then use the order of operations to simplify the expression. Common formulas include the area of a circle, A=πr2A = \pi r^2, and the volume of a cube, V=s3V = s^3.

Examples