OMPT drill — Algebra

OMPT exponents and roots practice: lesson and 10 problems

Q: What is the difference between a fractional exponent and a root on the OMPT?

They are the same object in two notations: a^1/n is the n-th root of a, and a^m/n is that root raised to the power m. The OMPT switches between them deliberately. Roots are easier to evaluate by eye, exponents are easier to simplify with the power rules, so translate to whichever form the question needs and translate back before you answer.

Q: Why does the OMPT keep asking whether -2 squared is 4 or -4?

Because the notation is a trap that costs marks. (-2)^2 squares the number -2 and gives 4. -2^2 squares 2 first and then applies the minus, giving -4, since an exponent binds tighter than a minus sign. The same care applies to sqrt(x^2), which equals |x|, not x. Pause for two seconds whenever a minus sign stands next to an exponent.

Power rules, fractional exponents, and rationalising denominators. The single most-reused skill on the whole test. Get it automatic before touching anything else. Work through all ten below before moving on; algebra slips compound into every other strand.

Tested inOMPT-A OMPT-B OMPT-C OMPT-D OMPT-E OMPT-F OMPT-G

OMPT exponents and roots: the lesson

What the OMPT tests in exponents and roots

Every power rule comes from one idea: an exponent counts repeated multiplication. $a^{3} \cdot a^{2}$ is five $a$ s multiplied together, so the exponents add. From that seed the rest grows: dividing subtracts, a power of a power multiplies, $a^{0} = 1$ and $a^{- n} = \frac{1}{a ^{n}}$ keep the pattern consistent, and $a^{1/ n}$ has to mean $n a$ because raising it to the $n$ must give $a$ back. The OMPT files this under the Numbers and Algebra chapters, and every variant from OMPT-A to OMPT-G draws on it, either as a direct question or as a step inside logarithms, exponential equations and differentiation.

The heavy sub-skills are four: simplifying an expression that mixes several rules at once, evaluating fractional and negative exponents without a calculator, simplifying and combining roots, and rationalising a denominator. A lighter fifth one, solving an equation by writing both sides as powers of the same base, turns up on OMPT-A, B, D and F. The wording is usually short: "Simplify", "Write as a single power of $x$ ", "Write in the form $a b$ ", "Evaluate". I put this topic first in every study plan I write, because if the rules are automatic four other chapters get easier at once.

Notation and answer format the OMPT expects

Exact forms, not decimals. Type $23$ as 2*sqrt(3) and $\frac{1}{8}$ as 1/8. A decimal is only accepted when the question says "round to".
Fractional and negative exponents need brackets: $x^{3/2}$ is x^(3/2) and $x^{- 2}$ is x^(-2). Without the brackets the platform reads x^3/2 as half of $x^{3}$ .
A root in a denominator is normally not accepted as final. $\frac{6}{3}$ must be entered as $23$ .
When the question says "as a power of $x$ ", give one power, $x^{1/2}$ , not $x$ . When it says "in root form", do the reverse.
For $x^{2}$ the honest answer is $∣ x ∣$ unless the question tells you $x \geq 0$ .

Worked example 1: a single power of x

Write $\frac{( 2 x ^{3} ) ^{2} \cdot x ^{- 5}}{4 x}$ as a single power of $x$ , for $x > 0$ .
Outer power first, applied to both factors inside: $(2 x^{3})^{2} = 4 x^{6}$ . The 2 gets squared as well as the $x^{3}$ .
Multiply like bases by adding exponents: $4 x^{6} \cdot x^{- 5} = 4 x$ .
Translate the root into exponent language: $x = x^{1/2}$ , so the denominator is $4 x^{1/2}$ .
Divide: $\frac{4 x}{4 x ^{1/2}} = x^{1 - 1/2} = x^{1/2}$ .
Check with $x = 4$ : the original is $\frac{( 2 \cdot 64 ) ^{2} \cdot 4 ^{- 5}}{4 \cdot 2} = \frac{16}{8} = 2$ , and $4^{1/2} = 2$ . A one-number test settles any simplification dispute.

Worked example 2: negative fractional exponents

Evaluate $3 2^{- 2/5}$ and $(\frac{1}{27})^{- 2/3}$ without a calculator.
Read the exponent in three parts: the sign says flip, the denominator says root, the numerator says power. Root first keeps the numbers small.
For $3 2^{- 2/5}$ : $532 = 2$ , then $2^{2} = 4$ , then the minus flips it: $\frac{1}{4}$ .
For $(\frac{1}{27})^{- 2/3}$ : the negative exponent turns $\frac{1}{27}$ into 27, leaving $2 7^{2/3} = 3^{2} = 9$ .
The trap is reading $3 2^{- 2/5}$ as $32 \cdot (- \frac{2}{5})$ . An exponent is never a multiplier, and the answer $- 12.8$ would be marked wrong with no partial credit.

The full lesson lists every rule with the condition attached to it, works through rationalising with a conjugate, matching bases in exponential equations, substitution when a power appears squared, and the parameter questions that OMPT-D and OMPT-F hide inside longer problems. It closes with the edge cases the examiners like and the eight slips I mark most often.

How each OMPT variant tests exponents and roots

The chapter weights are the official ones from the omptest.org learning-assessment pages. The question counts are my estimates of how much of that chapter this one topic takes up. The 21-question papers (OMPT-D and OMPT-F) ask long multi-part questions, so a topic there usually shows up as a step inside a bigger problem.

OMPT-A
52 questions
Testedroughly 4 to 6 of 52 questions
Official chapter: Numbers, 15% of the paper.
Short numeric or expression answers: simplify a product of powers, evaluate a fractional exponent, write a root sum in the form a*sqrt(b). Usually one equation solved by matching bases.
OMPT-B
64 questions
Testedroughly 4 to 6 of 64 questions
Official chapter: Numbers, 5% of the paper.
Same shapes as OMPT-A, plus power rules reused inside the differentiation and integration chapters, where a root has to be rewritten as a power before anything else happens.
OMPT-C
30 questions
Testedroughly 5 to 8 of 30 questions
Official chapter: Numbers, 35% of the paper.
The Numbers and Algebra chapters are half of this paper, so simplifying powers, evaluating roots and rationalising a denominator appear as direct short questions with exact answers.
OMPT-D
21 questions
Tested1 to 2 of the 21 long questions, usually as a step inside a bigger problem
Official chapter: Functions, equations and inequalities, 30% of the paper.
Rarely a bare simplification. Fractional exponents show up inside a derivative or a limit, and exponential equations with a parameter in the exponent appear as one part of a multi-part question.
OMPT-E
40 questions
Testedroughly 3 to 4 of 40 questions
Official chapter: Numbers, 10% of the paper.
Direct simplification and evaluation questions from the Numbers and Algebra chapters, with numeric answers. Exponent rules also feed the exponential functions chapter on this paper.
OMPT-F
21 questions
Tested1 to 2 of the 21 long questions, usually as a step inside a bigger problem
Official chapter: Algebra, 15% of the paper.
Power rules appear inside function and differentiation questions: rewrite a root as a power, simplify a quotient of powers, or solve an exponential equation as one step of a longer problem.
OMPT-G
26 questions
Testedroughly 1 to 3 of 26 questions, mostly inside other chapters
Official chapter: Functions, graphs and equations, 45% of the paper.
No standalone Numbers or Algebra chapter on this paper, so power rules are tested through exponential functions, geometric sequences and the physics and chemistry calculations that use them.

OMPT exponents and roots practice problems

10 original problems in the formats the OMPT uses: typed numeric answers, multiple choice, and spot-the-error. The last few are harder than the exam average on purpose. Solutions and the common mistake sit behind each problem.

Problem 1Numeric answer

Simplify and evaluate: $\frac{2 ^{3} \cdot 2 ^{5}}{2 ^{6}}$ .

Show the worked solution

Answer: 4

Multiplying powers with the same base means adding exponents: $2^{3} \cdot 2^{5} = 2^{8}$ . Dividing subtracts them: $2^{8 - 6} = 2^{2} = 4$ . I never compute the big numbers first — exponent rules do the work.

Problem 2Multiple choice

Simplify $(3 a^{2} b)^{3}$ .

A $27 a^{6} b^{3}$
B $9 a^{6} b^{3}$
C $27 a^{5} b^{3}$
D $3 a^{6} b^{3}$

Show the worked solution

Answer: A — $27 a^{6} b^{3}$

The cube applies to every factor inside the brackets: $3^{3} = 27$ , $(a^{2})^{3} = a^{6}$ (a power of a power multiplies exponents), and $b^{3}$ . Together: $27 a^{6} b^{3}$ .

Problem 3Numeric answer

Solve for $m$ : $5^{2 m - 1} = 125$ .

Show the worked solution

Answer: 2

I write both sides with base 5: $125 = 5^{3}$ , so $5^{2 m - 1} = 5^{3}$ . Equal bases mean equal exponents: $2 m - 1 = 3$ , so $m = 2$ . Checking: $5^{4 - 1} = 5^{3} = 125$ .

Problem 4Multiple choice

Simplify $50 + 18$ .

A $82$
B $68$
C $152$
D $417$

Show the worked solution

Answer: A — $82$

I pull out the largest square factors: $50 = 25 \cdot 2 = 52$ and $18 = 9 \cdot 2 = 32$ . Now they are like terms: $52 + 32 = 82$ .

Problem 5Spot the error

A student computes the hypotenuse of a right triangle with legs 3 and 4. Step 1: $c = 3^{2} + 4^{2}$ . Step 2: $c = 9 + 16$ . Step 3: $c = 3 + 4 = 7$ . Which step contains the error?

Show the worked solution

Answer: Step 2

Step 1 is the correct Pythagorean setup. Step 2 splits the root over the sum, and that is illegal: $a + b \neq = a + b$ . The right computation adds first, then takes the root: $9 + 16 = 25 = 5$ . The famous 3-4-5 triangle confirms it.

Problem 6Numeric answer

Evaluate $1 6^{3/4}$ .

Show the worked solution

Answer: 8

A fractional exponent is a root and a power combined: $1 6^{3/4} = (1 6^{1/4})^{3}$ . The fourth root of 16 is 2, and $2^{3} = 8$ . Taking the root first keeps the numbers small — cubing 16 first would give 4096 and only then a fourth root.

Problem 7Multiple choice

Simplify $\frac{( x ^{2} y ^{- 3} ) ^{2}}{x ^{- 1} y}$ for $x, y \neq = 0$ .

A $\frac{x ^{5}}{y ^{7}}$
B $\frac{x ^{3}}{y ^{7}}$
C $\frac{x ^{5}}{y ^{5}}$
D $\frac{x ^{3}}{y ^{5}}$

Show the worked solution

Answer: A — $\frac{x ^{5}}{y ^{7}}$

Outer power first: $(x^{2} y^{- 3})^{2} = x^{4} y^{- 6}$ , multiplying each exponent by 2. Now divide by $x^{- 1} y$ by subtracting exponents: $x^{4 - (- 1)} = x^{5}$ and $y^{- 6 - 1} = y^{- 7}$ . So the expression is $x^{5} y^{- 7} = \frac{x ^{5}}{y ^{7}}$ . Check with $x = 2, y = 1$ : the original is $\frac{16}{1/2} = 32 = 2^{5}$ .

Problem 8Numeric answer

Solve for $x$ : $3^{2 x} \cdot 9 = 2 7^{x - 1}$ .

Show the worked solution

Answer: 5

Base 3 on both sides. Left: $3^{2 x} \cdot 3^{2} = 3^{2 x + 2}$ . Right: $2 7^{x - 1} = (3^{3})^{x - 1} = 3^{3 x - 3}$ . Same base, so same exponent: $2 x + 2 = 3 x - 3$ , which gives $x = 5$ . Check: $3^{10} \cdot 9 = 59049 \cdot 9 = 531441$ and $2 7^{4} = 531441$ . They match.

Problem 9Numeric answer

The equation $4^{x} - 6 \cdot 2^{x} + 8 = 0$ has two real solutions. Give their sum.

Show the worked solution

Answer: 3

Since $4^{x} = (2^{x})^{2}$ , I substitute $t = 2^{x}$ with $t > 0$ : $t^{2} - 6 t + 8 = 0$ , so $(t - 2) (t - 4) = 0$ and $t = 2$ or $t = 4$ . Both positive, so both count. Back to $x$ : $2^{x} = 2$ gives $x = 1$ , $2^{x} = 4$ gives $x = 2$ . Sum: 3. Check: $4 - 12 + 8 = 0$ and $16 - 24 + 8 = 0$ .

Problem 10Numeric answer

For which value of $n$ is $2^{n} \cdot 3 4^{n} = 2^{14}$ ?

Show the worked solution

Answer: 12

Everything into base 2 with fractional exponents: $2^{n} = 2^{n /2}$ and $3 4^{n} = (2^{2 n})^{1/3} = 2^{2 n /3}$ . Multiplying adds the exponents: $\frac{n}{2} + \frac{2 n}{3} = \frac{3 n + 4 n}{6} = \frac{7 n}{6}$ . So $\frac{7 n}{6} = 14$ and $n = 12$ . Check: $2^{12} = 64$ , $3 4^{12} = 4^{4} = 256$ , and $64 \cdot 256 = 16384 = 2^{14}$ .

OMPT exponents and roots: questions students ask

Are exponents and roots on the OMPT-A?

Yes. The OMPT-A syllabus lists Numbers and Algebra as its first two chapters, and both lean on power rules: simplifying products and quotients of powers, negative and fractional exponents, and writing roots in simplest form. I would expect roughly 4 to 6 of the 52 questions to test these skills directly, and several more to use them as a step inside logarithm and differentiation questions.

Do I need a calculator for exponents and roots on the OMPT?

No, and you will not have one on the maths variants unless a specific question supplies it. That is why the questions use friendly numbers: 8, 27, 32, 81 and so on. Learn the small powers by heart (squares to 15, cubes to 5, powers of 2 to 1024) so that $3 2^{- 2/5}$ reads as "fifth root 2, squared 4, flipped 1/4" without any arithmetic.

How do I type a root or a fractional exponent as an OMPT answer?

Type roots as sqrt(2) and products as 2*sqrt(3). Type fractional or negative exponents with brackets, x^(3/2) and x^(-2), because x^3/2 means half of x cubed. Give exact forms unless the question says "round to", and do not leave a root in a denominator: 6/sqrt(3) should be entered as 2*sqrt(3). The platform checks the value, so an equivalent exact form is normally accepted.

What is the difference between a fractional exponent and a root on the OMPT?

They are the same object in two notations: $a^{1/ n}$ is the n-th root of a, and $a^{m / n}$ is that root raised to the power m. The OMPT switches between them deliberately. Roots are easier to evaluate by eye, exponents are easier to simplify with the power rules, so translate to whichever form the question needs and translate back before you answer.

How do I solve an exponential equation without logarithms on the OMPT?

Write both sides as powers of the same base and equate the exponents. For $4^{x + 1} = 82$ , both sides become powers of 2: 2x + 2 on the left, 7/2 on the right, so x = 3/4. If a power appears squared, as in $9^{x} - 4 \cdot 3^{x} + 3 = 0$ , substitute t for $3^{x}$ , solve the quadratic, and discard any negative t before converting back.

Why does the OMPT keep asking whether -2 squared is 4 or -4?

Because the notation is a trap that costs marks. $(- 2)^{2}$ squares the number -2 and gives 4. $- 2^{2}$ squares 2 first and then applies the minus, giving -4, since an exponent binds tighter than a minus sign. The same care applies to $x^{2}$ , which equals |x|, not x. Pause for two seconds whenever a minus sign stands next to an exponent.

Test hubs that cover exponents and roots:OMPT-A OMPT-B OMPT-C OMPT-D OMPT-E OMPT-F OMPT-G All 28 topics

OMPT exponents and roots practice: lesson and 10 problems

OMPT exponents and roots: the lesson

What the OMPT tests in exponents and roots

Notation and answer format the OMPT expects

Worked example 1: a single power of x

Worked example 2: negative fractional exponents

How each OMPT variant tests exponents and roots

OMPT exponents and roots practice problems

OMPT exponents and roots: questions students ask

Related OMPT topics and test hubs