5.2 Properties of Exponents and Scientific Notation

Exponents are powerful tools in algebra, allowing us to express repeated multiplication concisely. They come with handy rules that simplify complex calculations. These properties help us manipulate expressions and solve equations more efficiently.

Scientific notation takes exponents to the next level. It's a compact way to write very large or small numbers using powers of 10. This notation is crucial in science and engineering, making it easier to work with extreme values and perform calculations.

Properties of Exponents

Properties of exponents

• Product Rule multiplies the exponents when multiplying two powers with the same base ($x^3 \cdot x^4 = x^7$)
• Quotient Rule subtracts the exponents when dividing two powers with the same base ($\frac{x^5}{x^2} = x^3$)
• Power Rule multiplies the exponents when raising a power to another power ($(x^2)^3 = x^6$)
• Zero Exponent Rule states that any non-zero base raised to the power of 0 equals 1 ($5^0 = 1$)
• Negative Exponent Rule converts a negative exponent to its reciprocal with a positive exponent ($x^{-3} = \frac{1}{x^3}$)
• These properties are also known as the laws of exponents

Expressions with negative exponents

• Negative exponents represent the reciprocal of the base raised to the positive exponent ($x^{-a} = \frac{1}{x^a}$)
• When an expression with a negative exponent is in the denominator, move it to the numerator and change the exponent to positive ($\frac{1}{x^{-a}} = x^a$)
• When an expression with a negative exponent is in the numerator, move it to the denominator and change the exponent to positive ($x^{-a} = \frac{1}{x^a}$)
• Simplify expressions using the properties of exponents, combining like terms and canceling out common factors ($\frac{3x^{-2}y^3}{6x^2y^{-1}} = \frac{y^4}{2x^4}$)

Rational Exponents

• Rational exponents are exponents that can be expressed as fractions ($x^{\frac{m}{n}}$)
• The numerator represents the power, and the denominator represents the root
• For example, $x^{\frac{2}{3}}$ is equivalent to $\sqrt[3]{x^2}$
• Rational exponents follow the same properties as integer exponents

Scientific Notation

Standard vs scientific notation

• Scientific notation expresses very large or small numbers as a product of a number between 1 and 10 (inclusive) and a power of 10 ($a \times 10^n$, where $1 \leq |a| < 10$ and $n$ is an integer)
• The number between 1 and 10 is called the mantissa
• Standard notation is the usual way of writing numbers without powers of 10 (123,000 or 0.00456)
• To convert from standard to scientific notation:
  1. Move the decimal point to create a number between 1 and 10 (inclusive)
  2. Count the number of places moved and use that as the exponent ($n$)
     • If the decimal moved to the left, $n$ is positive (123,000 = $1.23 \times 10^5$)
     • If the decimal moved to the right, $n$ is negative (0.00456 = $4.56 \times 10^{-3}$)
• To convert from scientific to standard notation:
  1. Move the decimal point $n$ places
     • If $n$ is positive, move the decimal to the right ($1.23 \times 10^5$ = 123,000)
     • If $n$ is negative, move the decimal to the left ($4.56 \times 10^{-3}$ = 0.00456)
• When working with scientific notation, pay attention to significant figures to maintain precision

Order of Operations in Scientific Notation

• When performing calculations with numbers in scientific notation, follow the order of operations (PEMDAS)
• Perform operations on the mantissas and exponents separately
• Combine the results using the properties of exponents