Exponents and Powers – Key Vocabulary

Introduction

• Exponents (or powers) provide a compact way to write repeated multiplication.

  • Example of a very large quantity: mass of Earth \approx 5\,970\,000\,000\,000\,000\,000\,000\,000\,\text{kg}=5.97\times10^{24}\,\text{kg}.

  • Review of notation: 10^{24} is read “10 raised to the power 24”.

• Re-cap of positive exponents learned earlier:

  • 2^5 = 2\times2\times2\times2\times2.

  • For any non-zero integer a and positive integer m, a^m denotes m repeated factors of a.

• Guiding question raised in the text: “What is 2^{-2} equal to?”—leads into negative exponents.

Powers with Negative Exponents

• Pattern with base 10 illustrates meaning of negative powers:

  • 10^2 = 100

  • 10^1 = 10 (value becomes one-tenth of the previous step as the exponent drops by 1).

  • 10^0 = 1

  • 10^{-1}=\frac1{10}

  • 10^{-2}=\frac1{10^2}=\frac1{100}

  • 10^{-3}=\frac1{10^3}=\frac1{1000}

  • Continuing, 10^{-10}=\frac1{10^{10}}.

• Same logic shown with base 3:

  • 3^3=27,\;3^2=9,\;3^1=3,\;3^0=1.

  • Therefore 3^{-1}=\frac13,\;3^{-2}=\frac1{3^2}=\frac19,\;3^{-3}=\frac1{3^3}=\frac1{27}.

• General definition (valid for any non-zero integer a and positive integer m):

  • a^{-m}=\frac1{a^{m}}.

  • Interpretation: a^{-m} is the multiplicative inverse (reciprocal) of a^{m}.

• Quick practice (find the reciprocal):

  • 2^{-4}\;\Rightarrow\;\frac1{2^4}

  • 10^{-5}\;\Rightarrow\;\frac1{10^5}

  • 7^{-2}\;\Rightarrow\;\frac1{7^2}

  • 5^{-3}\;\Rightarrow\;\frac1{5^3}

  • 10^{-100}\;\Rightarrow\;\frac1{10^{100}} ("googolth" reciprocal).

Expanded Form of Whole and Decimal Numbers Using Exponents

• Whole numbers (review): 1425 = 1\times10^3 + 4\times10^2 + 2\times10^1 + 5\times10^0.

• Mixed decimal: 1425.36 can be decomposed as

  • 1\times10^3 + 4\times10^2 + 2\times10^1 + 5\times10^0 + 3\times10^{-1} + 6\times10^{-2}.

• Try-it-yourself: Expand 1025.63 or 1256.249 analogously, noting that each step right of the decimal corresponds to a negative exponent.

Laws of Exponents (Extended to All Integer Exponents)

• For any non-zero integer a, and any integers m,n:

  • Product rule: a^m\times a^n = a^{m+n}.

  • Quotient rule: \displaystyle \frac{a^m}{a^n}=a^{m-n}.

  • Power of a power: (a^m)^n = a^{mn}.

  • Product to a power (common exponent): a^m\times b^m = (ab)^m.

  • Quotient to a power: \displaystyle \frac{a^m}{b^m}=\left(\frac{a}{b}\right)^m.

  • Zero exponent: a^0 = 1 (provided a\neq0).

  • Reciprocal form: a^{-m}=\frac1{a^{m}} (already highlighted).

• Clarifications and caveats:

  • If the common base is neither 1 nor -1, equal powers imply equal exponents.

  • (-1)^p = 1 whenever p is even; (-1)^p=-1 when p is odd.

• Small proofs provided in text by substituting negative values to verify product rule.

Worked Examples (Computational Fluency)

• Evaluate simple negative powers:

  • 2^{-3}=\frac1{2^3}=\frac18.

  • \left(\frac13\right)^{-2}=\left(3^{-1}\right)^{-2}=3^{2}=9.

• Simplifications using laws:

  • ( -4)^5\times(-4)^{-10}=(-4)^{5-10}=(-4)^{-5}=\frac1{(-4)^5}.

  • 2^{5}\div 2^{-6}=2^{5-(-6)}=2^{11}.

  • Convert base: 4^{-3}\;(= (2^2)^{-3}) = 2^{2\times(-3)}=2^{-6}.

  • Combination example: \bigl(25\div 28\bigr)^{5}\times2^{-5}=(2^{5-8})^{5}\times2^{-5}=2^{-15}\times2^{-5}=2^{-20}=\frac1{2^{20}}.

• Solving for unknown exponent:

  • If ( -3)^{m+1}\times( -3)^5=( -3)^7 then m+1+5=7 \Rightarrow m=1.

Complex Fractions & Mixed Bases

• Illustrative simplifications (selected):

  • \left(\frac{1}{3}\right)^{-1}\left(\frac12\right)^{-1}\left(\frac14\right)^{-1}\left(\frac23\right)^{-2}\div\left(\frac23\right)^{-1} ultimately collapses to \frac1{16} after systematic use of product & quotient rules.

  • \bigl(\frac58\bigr)^{-7}\times\bigl(\frac85\bigr)^{-5}=(\frac58)^{-7+( -5)}=(\frac58)^{-12}=(\frac85)^{12} → intermediate steps illustrate base flipping.

Standard Form (Scientific Notation) for Very Large & Very Small Numbers

• Goal: write any number as a\times10^{k} where 1\le a<10 and k is an integer.

• Procedure for small numbers (decimal <1):

  1. Move decimal right until a single non-zero digit remains on the left.

  2. Number of moves =k; assign negative sign: 10^{-k}.

  • Example: 0.000007 = 7\times10^{-6}.

  • Example: 0.0016 = 1.6\times10^{-3} (thickness of paper).

• Procedure for large numbers (>10):

  1. Move decimal left until only one digit precedes the decimal.

  2. Number of moves =k; exponent is positive.

  • Example: 150\,000\,000\,000 = 1.5\times10^{11} (distance Earth-Sun).

• Converting back to usual form: multiply by 10^k or move decimal k places.

Real-World Data Expressed with Powers of 10

• Selected facts and their scientific notation (students instructed to complete table):

  • Distance Earth–Sun =1.496\times10^{11}\,\text{m}.

  • Speed of light \approx3.0\times10^{8}\,\text{m/s}.

  • Average Red Blood Cell diameter =7\times10^{-6}\,\text{m}.

  • Plant cell size =1.275\times10^{-5}\,\text{m}.

  • Thickness of human hair =(5\text{ to }10)\times10^{-3}\,\text{cm}.

  • Distance Earth–Moon =3.84467\times10^{8}\,\text{m}.

  • Radius of Sun =6.95\times10^{5}\,\text{km}.

  • Wire on a computer chip =3\times10^{-6}\,\text{m}.

• Students identify which are “very large” (positive exponent) vs. “very small” (negative exponent).

Comparing & Operating with Numbers in Standard Form

• Division (ratios):

  • Diameter Sun =1.4\times10^{9}; Earth =1.2756\times10^{7}.

  • \frac{1.4\times10^{9}}{1.2756\times10^{7}} \approx 100 → the Sun is ~100× wider.

• Ratio of tiny lengths:

  • Red blood cell vs. plant cell → \approx0.5 (red cell half the size).

• Addition / Subtraction: First match exponents.

  • Total mass Earth + Moon: 5.97\times10^{24}+7.35\times10^{22}=604.35\times10^{22}\,\text{kg}.

  • Distance Sun–Moon during solar eclipse:
    1.496\times10^{11} - 3.84\times10^{8} = 1.49216\times10^{11}\,\text{m} (after aligning exponents).

Key Take-Away Formulae & Facts

• Reciprocal rule: a^{-m}=\frac1{a^{m}}.

• Fundamental identities work for negative, positive, or zero exponents (except base 0 where undefined).

• Zero exponent gives 1 for any non-zero base.

• Even powers of -1 equal 1; odd powers equal -1.

• Scientific notation creates manageable numbers and preserves significant digits.

Suggested Practice (from Text)

• Evaluate: 3^{-2},\;( -4)^{-2},\;\left(\frac12\right)^{-5}.

• Simplify to positive exponents:

  • ( -4)^5\div( -4)^8

  • \left(\frac12\right)^{3} (convert if given negative exponent)

  • \left( -\frac34\right)^{5}\times\left(\frac34\right)^{3}, etc.

• Write in standard form: 0.000000564,\;0.0000021,\;21\,600\,000.

• Convert to ordinary form: 3.02\times10^{-6},\;4.5\times10^{4},\;5.8\times10^{12}.

• Real-world interpretation: compute total thickness of stack with 5 books of 20\,\text{mm} and 5 sheets of 0.016\,\text{mm}—requires adding numbers in different orders of magnitude.

Summary

• Negative exponents expand the exponent rules to include reciprocals, maintaining consistent algebraic patterns.

• Laws of exponents are universally valid for integer exponents and non-zero bases.

• Scientific notation (standard form) is an essential tool for science, engineering, and everyday contexts involving extreme scales.

• Mastery involves fluent conversion, comparison, and computation using both the algebraic laws and standard-form techniques.