Exponents of Whole Numbers - Definitive Study Guide

Fundamental Rules of Exponents for Whole Numbers

• Exponential notation represents repeated multiplication of a base number by itself.

• The general form of an exponential expression is given by:   $a^n$   where $a$ is the base and $n$ is the exponent or power.

• Zero Exponent Rule:

  • Any non-zero whole number raised to the power of zero is equal to 1.
  • The general mathematical expression is:     $a^0 = 1$
  • This holds true for any non-zero value of $a$.

• Identity Exponent Rule (Power of One):

  • Any whole number raised to the power of 1 is equal to the base itself.
  • The general mathematical expression is:     $a^1 = a$

• Powers of Ten Rule:

  • For a base of 10 raised to a whole number exponent $n$, the evaluated standard form consists of the digit 1 followed by exactly $n$ zeros.
  • The general mathematical expression is:     $10^n = 10^n$
  • For example, raising 10 to the third power yields 1 followed by 3 zeros:     $10^3 = 1000$

• Cube Exponents (Power of Three):

  • A base raised to the power of 3 is calculated by multiplying the base by itself three times:     $a^3 = a \times a \times a$

Conversion of Exponential Expressions to Standard Form

• Problem 1: Evaluation of $3^0$

  • Application of the Zero Exponent Rule.
  • Calculation:     $3^0 = 1$
  • Standard Form: 1

• Problem 2: Evaluation of $8^1$

  • Application of the Identity Exponent Rule.
  • Calculation:     $8^1 = 8$
  • Standard Form: 8

• Problem 3: Evaluation of $13^0$

  • Application of the Zero Exponent Rule.
  • Calculation:     $13^0 = 1$
  • Standard Form: 1

• Problem 4: Evaluation of $10^3$

  • Application of Powers of Ten Rule.
  • Multiplication expansion:     $10^3 = 10 \times 10 \times 10$
  • Calculation:     $10^3 = 1000$
  • Standard Form: 1000

• Problem 5: Evaluation of $15^3$

  • Multiplication expansion:     $15^3 = 15 \times 15 \times 15$
  • Step-by-step product:     $15 \times 15 = 225$$225 \times 15 = 3375$
  • Calculation:     $15^3 = 3375$
  • Standard Form: 3375

• Problem 6: Evaluation of $17^1$

  • Application of the Identity Exponent Rule.
  • Calculation:     $17^1 = 17$
  • Standard Form: 17

• Problem 7: Evaluation of $4^1$

  • Application of the Identity Exponent Rule.
  • Calculation:     $4^1 = 4$
  • Standard Form: 4

• Problem 8: Evaluation of $14^1$

  • Application of the Identity Exponent Rule.
  • Calculation:     $14^1 = 14$
  • Standard Form: 14

• Problem 9: Evaluation of $11^1$

  • Application of the Identity Exponent Rule.
  • Calculation:     $11^1 = 11$
  • Standard Form: 11

• Problem 10: Evaluation of $2^7$

  • Multiplication expansion:     $2^7 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$
  • Step-by-step intermediate products:     $2 \times 2 = 4$$4 \times 2 = 8$$8 \times 2 = 16$$16 \times 2 = 32$$32 \times 2 = 64$$64 \times 2 = 128$
  • Calculation:     $2^7 = 128$
  • Standard Form: 128

• Problem 11: Evaluation of $5^1$

  • Application of the Identity Exponent Rule.
  • Calculation:     $5^1 = 5$
  • Standard Form: 5

• Problem 12: Evaluation of $10^4$

  • Application of Powers of Ten Rule.
  • Multiplication expansion:     $10^4 = 10 \times 10 \times 10 \times 10$
  • Calculation:     $10^4 = 10000$
  • Standard Form: 10000

• Problem 13: Evaluation of $12^1$

  • Application of the Identity Exponent Rule.
  • Calculation:     $12^1 = 12$
  • Standard Form: 12

• Problem 14: Evaluation of $17^3$

  • Multiplication expansion:     $17^3 = 17 \times 17 \times 17$
  • Step-by-step product:     $17 \times 17 = 289$$289 \times 17 = 4913$
  • Calculation:     $17^3 = 4913$
  • Standard Form: 4913

• Problem 15: Evaluation of $25^1$

  • Application of the Identity Exponent Rule.
  • Calculation:     $25^1 = 25$
  • Standard Form: 25

• Problem 16: Evaluation of $10^4$

  • Application of Powers of Ten Rule.
  • Multiplication expansion:     $10^4 = 10 \times 10 \times 10 \times 10$
  • Calculation:     $10^4 = 10000$
  • Standard Form: 10000

• Problem 17: Evaluation of $2^0$

  • Application of the Zero Exponent Rule.
  • Calculation:     $2^0 = 1$
  • Standard Form: 1