Simplify expressions involving integer exponents using the laws of exponents, and represent and operate with very large or very small numbers in scientific notation (A.EO.2).

What this topic is asking

A.EO.2 asks you to simplify expressions with integer exponents using the laws of exponents, and to convert between standard form and scientific notation for very large or very small numbers. On the Virginia Algebra I SOL these appear as multiple choice, fill-in-the-blank (type the simplified expression), and matching (pair an expression with its simplified form). The laws of exponents are on the formula sheet, but you must apply them fluently.

The laws of exponents

An exponent counts repeated multiplication: $x^4 = x \cdot x \cdot x \cdot x$. The laws follow from this directly.

• Product of powers. $x^a \cdot x^b = x^{a+b}$. Multiplying like bases adds exponents: $x^3 \cdot x^5 = x^8$.
• Quotient of powers. $\dfrac{x^a}{x^b} = x^{a-b}$. Dividing like bases subtracts exponents: $\dfrac{x^7}{x^2} = x^5$.
• Power of a power. $(x^a)^b = x^{ab}$. Raising a power to a power multiplies exponents: $(x^3)^4 = x^{12}$.
• Power of a product. $(xy)^a = x^a y^a$, and $\left(\dfrac{x}{y}\right)^a = \dfrac{x^a}{y^a}$. The exponent reaches every factor.
• Zero exponent. $x^0 = 1$ for any nonzero $x$.
• Negative exponent. $x^{-n} = \dfrac{1}{x^n}$. A negative exponent does not make a number negative; it moves the factor to the other side of the fraction bar.

Scientific notation

Scientific notation writes a number as

The coefficient $a$ has exactly one nonzero digit before the decimal point. The exponent $n$ records the size: positive for numbers $\ge 10$ (the decimal moves left to make $a$), and negative for numbers between $0$ and $1$ (the decimal moves right). For example $93{,}000{,}000 = 9.3 \times 10^7$ and $0.00052 = 5.2 \times 10^{-4}$.

To convert back to standard form, move the decimal the number of places given by the exponent: right for a positive exponent, left for a negative one. $3.6 \times 10^5 = 360{,}000$ and $7.1 \times 10^{-3} = 0.0071$.

Operating with scientific notation

To multiply, multiply the coefficients and add the exponents: $(3 \times 10^4)(2 \times 10^5) = 6 \times 10^9$. To divide, divide the coefficients and subtract the exponents. If the new coefficient falls outside $[1, 10)$, renormalize: $(5 \times 10^3)(4 \times 10^2) = 20 \times 10^5 = 2 \times 10^6$, shifting the extra factor of $10$ into the exponent. The Desmos calculator handles these directly, but you should be able to reason through them by hand.

Why a negative exponent is a reciprocal, not a negative number

The pattern that defines a negative exponent comes from the quotient law. Dividing $x^3$ by $x^5$ gives $x^{3-5} = x^{-2}$ by subtraction, but it also equals $\dfrac{x^3}{x^5} = \dfrac{1}{x^2}$ by cancellation. For both to agree, $x^{-2}$ must mean $\dfrac{1}{x^2}$. So a negative exponent is the law of quotients telling you the base belongs in the denominator. It never changes the sign of the value: $2^{-3} = \frac{1}{8}$, a positive number. Likewise the zero exponent comes from $\dfrac{x^a}{x^a} = x^0$ and $\dfrac{x^a}{x^a} = 1$, forcing $x^0 = 1$. The "rules" are not arbitrary; they are what keeps the laws consistent.

How the SOL examines this topic

• Fill-in-the-blank. Simplify an exponential expression and type it with positive exponents.
• Multiple choice. Convert to or from scientific notation, or pick the correctly simplified expression.
• Matching / drag-and-drop. Pair expressions with equivalent simplified forms, or order steps.

Try this

Q1. Simplify $(3x^2)^3$. [1 point]

• Cue. $3^3 x^{2 \cdot 3} = 27x^6$.

Q2. Write $0.00018$ in scientific notation. [1 point]

• Cue. $1.8 \times 10^{-4}$ (decimal moves $4$ right; small number, negative exponent).