2.2 Exponents & Scientific Notation

2.2 Exponents & Scientific Notation

Exponential notation and scientific notation provide compact methods for representing repeated multiplication and working with extremely large or small numbers. Mastering the algebraic rules governing powers, rational exponents, and scientific calculations is a fundamental requirement for the CLEP College Algebra exam.

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Fundamental Laws of Integer Exponents

For any real numbers $a, b$ as variable bases and any integers $m, n$ as powers (assuming non-zero bases when denominators or negative powers appear):

Parentheses & Negative Sign Pitfall:

• $(-4)^2 = (-4)(-4) = +16$ (The base is $-4$).
• $-4^2 = -(4 \cdot 4) = -16$ (The base is $4$; negative sign is applied after squaring).

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Multi-Step Simplification of Complex Expressions

Simplifying expressions involving multiple variables and negative exponents requires systematically applying exponent rules until:

1. Each variable base appears only once.
2. No negative exponents remain.
3. All coefficients are simplified.

Worked Example: Complex Exponential Simplification

Problem: Completely simplify the expression and write the answer using only positive exponents: $\left( \frac{24 x^{-5} y^7 z^{-3}}{3 x^3 y^{-2} z^5} \right)^{-2}$

Step-by-Step Solution:

1. Simplify the numerical coefficients inside parentheses: $\frac{24}{3} = 8$

2. Apply Quotient Rule $\left(\frac{a^m}{a^n} = a^{m-n}\right)$ to inside variables:

   • For $x$: $\frac{x^{-5}}{x^3} = x^{-5 - 3} = x^{-8}$.
   • For $y$: $\frac{y^7}{y^{-2}} = y^{7 - (-2)} = y^{7 + 2} = y^9$.
   • For $z$: $\frac{z^{-3}}{z^5} = z^{-3 - 5} = z^{-8}$.

   Inner simplified expression: $\left( 8 x^{-8} y^9 z^{-8} \right)^{-2}$.

3. Distribute outer exponent $(-2)$ to all inner factors: $8^{-2} \cdot (x^{-8})^{-2} \cdot (y^9)^{-2} \cdot (z^{-8})^{-2}$

4. Apply Power of a Power Rule $\left((a^m)^n = a^{mn}\right)$:

   • Numerical factor: $8^{-2} = \frac{1}{8^2} = \frac{1}{64}$.
   • $x$ term: $(x^{-8})^{-2} = x^{(-8)(-2)} = x^{16}$.
   • $y$ term: $(y^9)^{-2} = y^{(9)(-2)} = y^{-18}$.
   • $z$ term: $(z^{-8})^{-2} = z^{(-8)(-2)} = z^{16}$.

5. Rewrite with positive exponents: $\frac{x^{16} z^{16}}{64 y^{18}}$

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Rational Exponents & Radical Conversions

A rational exponent is an exponent expressed as a fraction $\frac{m}{n}$, where $m \in \mathbb{Z}$ and $n \in \mathbb{Z}^+$ ($n > 0$).

$a^{m/n} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m$

• The denominator $n$ represents the index of the radical root.
• The numerator $m$ represents the power to which the base or root is raised.

Equivalence Table: Radicals vs. Rational Exponents

Worked Example: Rational Exponent Evaluation

Problem: Evaluate the numerical value of $(-125)^{-2/3} + (81)^{3/4}$.

Step-by-Step Solution:

1. Evaluate $(-125)^{-2/3}$:

   • Apply negative exponent: $\frac{1}{(-125)^{2/3}}$.
   • Take the cube root first: $\sqrt[3]{-125} = -5$.
   • Square the result: $(-5)^2 = 25$.
   • Result: $\frac{1}{25}$.

2. Evaluate $(81)^{3/4}$:

   • Take the 4th root first: $\sqrt[4]{81} = 3$.
   • Cube the result: $3^3 = 27$.

3. Combine terms via addition: $\frac{1}{25} + 27 = \frac{1}{25} + \frac{675}{25} = \frac{676}{25} = 27.04$

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Scientific Notation Operations

A real number is written in scientific notation when expressed as: $a \times 10^n$ where $1 \le |a| < 10$ and $n$ is an integer ($\mathbb{Z}$).

• If $n > 0$, the number is large ($> 10$).
• If $n < 0$, the number is small ($0 < |x| < 1$).
• If $n = 0$, $10^0 = 1$, representing numbers between 1 and 9.999...

Arithmetic Rules in Scientific Notation

• Multiplication: $(a \times 10^m)(b \times 10^n) = (a \cdot b) \times 10^{m+n}$
• Division: $\frac{a \times 10^m}{b \times 10^n} = \left(\frac{a}{b}\right) \times 10^{m-n}$

Worked Example: Scientific Notation Calculation

Problem: Compute the quotient in standard scientific notation: $\frac{(7.2 \times 10^{11})(4.5 \times 10^{-4})}{1.8 \times 10^{-5}}$

Step-by-Step Solution:

1. Multiply the numerators:

   • Coefficients: $7.2 \times 4.5 = 32.4$.
   • Powers of 10: $10^{11} \times 10^{-4} = 10^{11 + (-4)} = 10^7$.
   • Numerator product: $32.4 \times 10^7$.

2. Divide by denominator:

   • Coefficients: $\frac{32.4}{1.8} = 18$.
   • Powers of 10: $\frac{10^7}{10^{-5}} = 10^{7 - (-5)} = 10^{12}$.
   • Quotient: $18 \times 10^{12}$.

3. Adjust coefficient to standard form ($1 \le a < 10$):

   • Rewrite $18 = 1.8 \times 10^1$.
   • $(1.8 \times 10^1) \times 10^{12} = 1.8 \times 10^{13}$.