6.2 Exponents, Radicals, and Scientific Notation
Key Takeaways
- Exponents accounts for 2 to 3 of the 20 QAS questions, roughly 10 to 15 percent of the test.
- Multiplying like bases adds exponents and dividing subtracts them; raising a power to a power multiplies them.
- A negative exponent means the reciprocal, so x to the negative n equals 1 over x to the n.
- A fractional exponent is a radical: x to the m over n is the nth root of x to the m.
- Any nonzero base raised to the zero power equals 1, which is the most commonly missed single fact in this area.
A Named QAS Content Area
The Quantitative Reasoning, Algebra, and Statistics test lists ten content areas. Exponents is one of them, carrying 2 to 3 of the 20 questions:
Exponents: Calculating with exponents, radicals, fractional exponents, and applying scientific notation.
Four distinct competencies in one line. Each is treated below.
The Exponent Laws
| Law | Rule | Example |
|---|---|---|
| Product | $x^a \cdot x^b = x^{a+b}$ | $x^3 \cdot x^5 = x^8$ |
| Quotient | $x^a \div x^b = x^{a-b}$ | $x^7 \div x^2 = x^5$ |
| Power of a power | $(x^a)^b = x^{ab}$ | $(x^3)^4 = x^{12}$ |
| Power of a product | $(xy)^a = x^a y^a$ | $(2x)^3 = 8x^3$ |
| Power of a quotient | $(x/y)^a = x^a / y^a$ | $(x/3)^2 = x^2/9$ |
| Zero exponent | $x^0 = 1$, $x \neq 0$ | $47^0 = 1$ |
| Negative exponent | $x^{-a} = 1/x^a$ | $2^{-3} = 1/8$ |
The Two Most Common Errors
Adding exponents when multiplying bases, not powers. The product rule applies only to like bases:
$x^3 \cdot y^4$ does not simplify. Different bases. $2^3 \cdot 2^4 = 2^7 = 128$. ✓
Assuming $x^0 = 0$. It equals 1. This follows from the quotient rule: $x^3 \div x^3 = x^0$, and any nonzero quantity divided by itself is 1.
Power of a Product Includes the Coefficient
$(3x^2)^3 = 3^3 \cdot x^{2 \cdot 3} = 27x^6$
Students routinely produce $3x^6$, forgetting that the exponent applies to the coefficient too.
Negative Exponents
A negative exponent signals a reciprocal, not a negative value:
$2^{-3} = \dfrac{1}{2^3} = \dfrac{1}{8}$
$2^{-3}$ is positive. It is small, not negative.
Moving a factor across the fraction bar flips the sign of its exponent:
$\dfrac{x^{-2}}{y^{-3}} = \dfrac{y^3}{x^2}$
Fractional Exponents and Radicals
The bridge between exponents and roots:
$x^{1/n} = \sqrt[n]{x}$ and $x^{m/n} = \sqrt[n]{x^m} = \left(\sqrt[n]{x}\right)^m$
The denominator is the root; the numerator is the power.
$8^{2/3} = \left(\sqrt[3]{8}\right)^2 = 2^2 = 4$
Taking the root first keeps the numbers small — always preferable to computing $8^2 = 64$ and then finding $\sqrt[3]{64}$.
$16^{3/4} = \left(\sqrt[4]{16}\right)^3 = 2^3 = 8$
Simplifying Radicals
Extract perfect-square factors:
$\sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2}$
Find the largest perfect square that divides the radicand. Using $\sqrt{4 \cdot 18} = 2\sqrt{18}$ is correct but not fully simplified, since 18 still contains a factor of 9.
Perfect squares to recognize on sight: 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144.
Radical Arithmetic
- Multiply: $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$, so $\sqrt{6} \cdot \sqrt{10} = \sqrt{60} = 2\sqrt{15}$
- Add/subtract: only like radicals combine, exactly as with like terms. $3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5}$, but $\sqrt{2} + \sqrt{3}$ does not simplify.
A frequent error: $\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}$. Check with numbers: $\sqrt{9 + 16} = \sqrt{25} = 5$, but $\sqrt{9} + \sqrt{16} = 3 + 4 = 7$. Not equal.
Rationalizing a Denominator
$\dfrac{6}{\sqrt{3}} = \dfrac{6}{\sqrt{3}} \cdot \dfrac{\sqrt{3}}{\sqrt{3}} = \dfrac{6\sqrt{3}}{3} = 2\sqrt{3}$
Order of Operations With Exponents
Exponents sit above multiplication and division in the order of operations, which produces results that surprise students working quickly.
$3 \cdot 2^4 = 3 \cdot 16 = 48$, not $6^4$.
The exponent binds only to the 2. To raise the whole product, parentheses are required: $(3 \cdot 2)^4 = 6^4 = 1296$.
The same principle governs signs:
| Expression | Value | Why |
|---|---|---|
| $(-4)^2$ | 16 | The parentheses include the sign in the base |
| $-4^2$ | −16 | The exponent binds to 4; the negative applies afterward |
| $(-2)^3$ | −8 | An odd power preserves a negative base |
| $-2^3$ | −8 | Same value here, but for a different reason |
That third and fourth row coincide by accident. With an even exponent they diverge sharply, which is exactly where test items are built.
Negative bases follow a parity rule: a negative base raised to an even power is positive; raised to an odd power it stays negative.
Exponents Inside Fractions
When a quotient of like bases has the larger exponent in the denominator, the result is a negative exponent — that is, a fraction:
Both forms are correct, but answer choices usually prefer the form with no negative exponents, so convert before matching.
Scientific Notation
A number in scientific notation is written $a \times 10^n$ where $1 \le |a| < 10$ and $n$ is an integer.
$4{,}730{,}000 = 4.73 \times 10^6$ $0.000082 = 8.2 \times 10^{-5}$
The exponent counts decimal-point moves: right-to-left moves give a positive exponent; left-to-right moves give a negative one.
Arithmetic in Scientific Notation
Multiplication — multiply the coefficients, add the exponents:
$(3.0 \times 10^4)(2.0 \times 10^3) = 6.0 \times 10^7$
Division — divide the coefficients, subtract the exponents:
$\dfrac{8.4 \times 10^9}{2.1 \times 10^4} = 4.0 \times 10^5$
Renormalizing. If the coefficient falls outside $[1, 10)$, adjust:
$(5.0 \times 10^3)(4.0 \times 10^6) = 20.0 \times 10^9 = 2.0 \times 10^{10}$
Addition and subtraction require matching exponents first:
$3.2 \times 10^5 + 4.0 \times 10^4 = 3.2 \times 10^5 + 0.4 \times 10^5 = 3.6 \times 10^5$
This is the step most often skipped, and it produces answers off by an order of magnitude — which is exactly what the distractors are built to catch.
Simplify $(2x^3)^4$.
Evaluate $27^{2/3}$.
Compute $2.4 \times 10^6 + 3.0 \times 10^5$ and express the result in scientific notation.