5.2 Properties of Exponents and Scientific Notation

Properties of Exponents

Properties of Exponents

Every exponent rule comes from one idea: exponents count how many times a base is multiplied by itself. The rules below let you simplify expressions without expanding everything out.

• Product Rule — when multiplying powers with the same base, add the exponents: $x^3 \cdot x^4 = x^{3+4} = x^7$
• Quotient Rule — when dividing powers with the same base, subtract the exponents: $\frac{x^5}{x^2} = x^{5-2} = x^3$
• Power Rule — when raising a power to another power, multiply the exponents: $(x^2)^3 = x^{2 \cdot 3} = x^6$
• Zero Exponent Rule — any non-zero base raised to the 0 power equals 1: $5^0 = 1$
• Negative Exponent Rule — a negative exponent means "take the reciprocal": $x^{-3} = \frac{1}{x^3}$

A common mistake with the Product Rule: students sometimes multiply the exponents instead of adding them. Remember, $x^3 \cdot x^4 = x^7$, not $x^{12}$. Multiplying exponents only happens with the Power Rule.

Expressions with Negative Exponents

Negative exponents don't make a number negative. They flip the base to the other side of a fraction bar and then become positive.

• A negative exponent in the numerator moves the factor to the denominator: $x^{-a} = \frac{1}{x^a}$
• A negative exponent in the denominator moves the factor to the numerator: $\frac{1}{x^{-a}} = x^a$

To simplify an expression with negative exponents, move every factor with a negative exponent across the fraction bar, then apply the other exponent rules as usual.

Example: Simplify $\frac{3x^{-2}y^3}{6x^2y^{-1}}$

1. Move $x^{-2}$ from the numerator down and $y^{-1}$ from the denominator up: $\frac{3 \cdot y^3 \cdot y^1}{6 \cdot x^2 \cdot x^2}$
2. Combine like bases using the Product Rule: $\frac{3y^4}{6x^4}$
3. Simplify the coefficient: $\frac{y^4}{2x^4}$

Rational Exponents

Rational exponents are exponents written as fractions, like $x^{\frac{m}{n}}$. They connect exponents to roots:

• The denominator of the fraction is the root (index).
• The numerator is the power.

So $x^{\frac{m}{n}} = \sqrt[n]{x^m}$. For example, $x^{\frac{2}{3}} = \sqrt[3]{x^2}$.

All the same exponent properties apply to rational exponents. That's what makes them useful: you can use the Product Rule, Power Rule, etc. on roots without switching to radical notation.

Scientific Notation

Standard vs. Scientific Notation

Scientific notation expresses numbers as $a \times 10^n$, where $1 \leq |a| < 10$ and $n$ is an integer. The value $a$ is sometimes called the coefficient (or mantissa), and $n$ tells you the order of magnitude. Standard notation is just the regular way of writing numbers, like 123,000 or 0.00456.

Converting standard → scientific notation:

1. Move the decimal point until you have a number between 1 and 10.
2. Count how many places you moved the decimal. That count becomes $n$.
   • Decimal moved left → $n$ is positive (the original number is large): $123{,}000 = 1.23 \times 10^5$
   • Decimal moved right → $n$ is negative (the original number is small): $0.00456 = 4.56 \times 10^{-3}$

Converting scientific → standard notation:

1. Look at the exponent $n$.
   • If $n$ is positive, move the decimal to the right $n$ places: $1.23 \times 10^5 = 123{,}000$
   • If $n$ is negative, move the decimal to the left $|n|$ places: $4.56 \times 10^{-3} = 0.00456$

Operations with Scientific Notation

When multiplying or dividing numbers in scientific notation, handle the coefficients and the powers of 10 separately, then combine.

Multiplication: Multiply the coefficients and add the exponents.

$\quad (3 \times 10^4)(2 \times 10^3) = (3 \cdot 2) \times 10^{4+3} = 6 \times 10^7$

Division: Divide the coefficients and subtract the exponents.

$\quad \frac{8 \times 10^6}{4 \times 10^2} = \frac{8}{4} \times 10^{6-2} = 2 \times 10^4$

If your result's coefficient falls outside the range 1 to 10, adjust it. For instance, $12 \times 10^3$ becomes $1.2 \times 10^4$.

For addition and subtraction, the exponents must match first. Rewrite one of the numbers so both share the same power of 10, then add or subtract the coefficients.