Learn on PengiSaxon Math, Course 2Chapter 6: Lessons 51-60, Investigation 6

Lesson 57: Negative Exponents, Scientific Notation for Small Numbers

In this Grade 7 Saxon Math Course 2 lesson, students learn how negative exponents work using the rule a⁻ⁿ = 1/aⁿ and how to simplify expressions like 3⁻² and 10⁻³. Students also extend their understanding of scientific notation to small numbers, practicing how to convert between scientific notation with negative exponents and standard decimal form by shifting the decimal point to the left.

Section 1

📘 Negative Exponents and Scientific Notation

New Concept

This course builds your mathematical toolkit, showing how numbers, operations, and variables work together to solve problems systematically and efficiently.

What’s next

To start, we'll explore negative exponents and how they help us write very small numbers using scientific notation, with worked examples and practice problems.

Section 2

Negative Exponents

Property

For any non-zero number aa and any integer nn, an=1ana^{-n} = \frac{1}{a^n}.

Examples

32=132=193^{-2} = \frac{1}{3^2} = \frac{1}{9}
104=1104=11000010^{-4} = \frac{1}{10^4} = \frac{1}{10000}
53=153=11255^{-3} = \frac{1}{5^3} = \frac{1}{125}

Explanation

A negative exponent is a secret signal to flip the number! Instead of multiplying by the base, you are dividing. This cool trick transforms a power into its reciprocal, turning it into a fraction. It’s like a shrinking ray for numbers, perfect for expressing tiny values without a bunch of decimals.

Section 3

The Zero Exponent

Property

If a number aa is not zero, then a0=1a^0 = 1.

Examples

1500=1150^0 = 1
(8)0=1(-8)^0 = 1
(27)0=1(\frac{2}{7})^0 = 1

Explanation

Think of the zero exponent as the great equalizer in math. Any non-zero number, whether it's gigantic or super small, gets zapped down to exactly 1 when raised to the power of zero. It is the ultimate reset button for bases, making it a handy shortcut for simplifying complex expressions.

Section 4

Scientific Notation for Small Numbers

Property

To convert a number from scientific notation with a negative exponent to standard form, shift the decimal point to the left by the number of places indicated by the exponent.

Examples

Write in standard form: 5.12×107=0.0000005125.12 \times 10^{-7} = 0.000000512
Write in scientific notation: 0.000071=7.1×1050.000071 = 7.1 \times 10^{-5}
Write in standard form: 2.9×104=0.000292.9 \times 10^{-4} = 0.00029

Explanation

That little negative sign on the exponent is a command to make the number smaller. If you see 10N10^{-N}, it means you need to hop the decimal point N places to the left, adding zeros as placeholders. It’s how scientists write tiny measurements, like the width of a hair, without tons of zeros.

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 6: Lessons 51-60, Investigation 6

  1. Lesson 1

    Lesson 51: Scientific Notation for Large Numbers

  2. Lesson 2

    Lesson 52: Order of Operations

  3. Lesson 3

    Lesson 53: Ratio Word Problems

  4. Lesson 4

    Lesson 54: Rate Word Problems

  5. Lesson 5

    Lesson 55: Average and Rate Problems with Multiple Steps

  6. Lesson 6

    Lesson 56: Plotting Functions

  7. Lesson 7Current

    Lesson 57: Negative Exponents, Scientific Notation for Small Numbers

  8. Lesson 8

    Lesson 58: Symmetry

  9. Lesson 9

    Lesson 59: Adding Integers on the Number Line

  10. Lesson 10

    Lesson 60: Fractional Part of a Number, Part 1, Percent of a Number, Part 1

  11. Lesson 11

    Investigation 6: Classifying Quadrilaterals

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

📘 Negative Exponents and Scientific Notation

New Concept

This course builds your mathematical toolkit, showing how numbers, operations, and variables work together to solve problems systematically and efficiently.

What’s next

To start, we'll explore negative exponents and how they help us write very small numbers using scientific notation, with worked examples and practice problems.

Section 2

Negative Exponents

Property

For any non-zero number aa and any integer nn, an=1ana^{-n} = \frac{1}{a^n}.

Examples

32=132=193^{-2} = \frac{1}{3^2} = \frac{1}{9}
104=1104=11000010^{-4} = \frac{1}{10^4} = \frac{1}{10000}
53=153=11255^{-3} = \frac{1}{5^3} = \frac{1}{125}

Explanation

A negative exponent is a secret signal to flip the number! Instead of multiplying by the base, you are dividing. This cool trick transforms a power into its reciprocal, turning it into a fraction. It’s like a shrinking ray for numbers, perfect for expressing tiny values without a bunch of decimals.

Section 3

The Zero Exponent

Property

If a number aa is not zero, then a0=1a^0 = 1.

Examples

1500=1150^0 = 1
(8)0=1(-8)^0 = 1
(27)0=1(\frac{2}{7})^0 = 1

Explanation

Think of the zero exponent as the great equalizer in math. Any non-zero number, whether it's gigantic or super small, gets zapped down to exactly 1 when raised to the power of zero. It is the ultimate reset button for bases, making it a handy shortcut for simplifying complex expressions.

Section 4

Scientific Notation for Small Numbers

Property

To convert a number from scientific notation with a negative exponent to standard form, shift the decimal point to the left by the number of places indicated by the exponent.

Examples

Write in standard form: 5.12×107=0.0000005125.12 \times 10^{-7} = 0.000000512
Write in scientific notation: 0.000071=7.1×1050.000071 = 7.1 \times 10^{-5}
Write in standard form: 2.9×104=0.000292.9 \times 10^{-4} = 0.00029

Explanation

That little negative sign on the exponent is a command to make the number smaller. If you see 10N10^{-N}, it means you need to hop the decimal point N places to the left, adding zeros as placeholders. It’s how scientists write tiny measurements, like the width of a hair, without tons of zeros.

Book overview

Jump across lessons in the current chapter without opening the full course modal.

Continue this chapter

Chapter 6: Lessons 51-60, Investigation 6

  1. Lesson 1

    Lesson 51: Scientific Notation for Large Numbers

  2. Lesson 2

    Lesson 52: Order of Operations

  3. Lesson 3

    Lesson 53: Ratio Word Problems

  4. Lesson 4

    Lesson 54: Rate Word Problems

  5. Lesson 5

    Lesson 55: Average and Rate Problems with Multiple Steps

  6. Lesson 6

    Lesson 56: Plotting Functions

  7. Lesson 7Current

    Lesson 57: Negative Exponents, Scientific Notation for Small Numbers

  8. Lesson 8

    Lesson 58: Symmetry

  9. Lesson 9

    Lesson 59: Adding Integers on the Number Line

  10. Lesson 10

    Lesson 60: Fractional Part of a Number, Part 1, Percent of a Number, Part 1

  11. Lesson 11

    Investigation 6: Classifying Quadrilaterals