16x^{12}. The power-of-a-product rule applies the outer exponent to every factor inside the parentheses, including the coefficient. So 2^4 = 16 and (x^3)^4 = x^{3 \times 4} = x^{12}, giving 16x^{12}. The most frequent error is leaving the coefficient untouched to get 2x^{12}, and the second most frequent is adding the exponents instead of multiplying them to get x^7.

In a fractional exponent the denominator gives the root and the numerator gives the power. Taking the cube root first keeps the arithmetic small: the cube root of 27 is 3, and squaring gives 9. Computing the power first would require the cube root of 729, which is also 9 but is far harder without a calculator. The answer 729 results from treating the exponent as if it were 3 rather than 2/3.

Compute 2.4 \times 10^6 + 3.0 \times 10^5 and express the result in scientific notation.

2.7 \times 10^6. Addition in scientific notation requires matching exponents before the coefficients can be combined. Rewriting the second term as 0.3 \times 10^6 gives 2.4 \times 10^6 + 0.3 \times 10^6 = 2.7 \times 10^6. Adding the coefficients directly without aligning exponents produces 5.4, and adding the exponents as if this were multiplication produces the power of 11 — both are the standard distractors for this item type.

Exponents, Radicals, and Scientific Notation | Free Guide 2026

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.

(-3)^4 = 81 \qquad (-3)^5 = -243

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:

\frac{x^2}{x^6} = x^{2-6} = x^{-4} = \frac{1}{x^4}

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.

ACCUPLACER

6.2 Exponents, Radicals, and Scientific Notation

Key Takeaways