Study guide
2.4 Radicals, Rational Expressions, and Scientific Notation
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1 Introduction & Exam Overview
1.1 AdUCET Overview, Format, and Admission Process 1.2 Test-Taking and Time-Management Strategies
2 Chapter 2: Mathematics I — Numbers, Algebra, and Functions
2.1 Number Sense, Operations, and Fraction–Decimal–Percent Conversions 2.2 Algebra, Polynomials, and Linear/Quadratic Equations 2.3 Inequalities, Absolute Value, and Systems of Equations 2.4 Radicals, Rational Expressions, and Scientific Notation 2.5 Coordinate Geometry, Lines, and Functions
3 Chapter 3: Mathematics II — Geometry, Measurement, and Trigonometry
3.1 Plane Geometry: Angles, Triangles, Polygons, and Circles 3.2 Solid Mensuration, Measurement, and Unit Conversion 3.3 Right-Triangle Trigonometry and Applications
4 Chapter 4: Mathematics III — Sequences, Proportion, and Data
4.1 Arithmetic and Geometric Sequences and Series 4.2 Ratio, Proportion, Percentage, Work, and Mixture Problems 4.3 Descriptive Statistics and Data Interpretation 4.4 Probability: Rules, Compound Events, and Conditional Probability 4.5 Counting Principles: Permutations and Combinations
5 Chapter 5: English I — Reading Comprehension and Vocabulary
5.1 Reading Comprehension: Main Idea, Purpose, and Passage Structure 5.2 Inference, Supporting Detail, and Author's Tone 5.3 Word Roots, Prefixes, and Suffixes 5.4 Vocabulary in Context, Synonyms, Antonyms, and Confusables
6 Chapter 6: English II — Grammar, Usage, and Written Expression
6.1 Subject-Verb Agreement and Verb Tenses 6.2 Pronouns, Modifiers, and Parallel Structure 6.3 Sentence Boundaries, Clauses, and Punctuation 6.4 Sentence Correction, Redundancy, and Idiomatic Usage 6.5 Verbal Analogies and Relationship Bridges 6.6 Paragraph Organization, Coherence, and Transitions
7 Chapter 7: General Science I — Biology and Chemistry
7.1 Cell Biology, Transport, and Cellular Energetics 7.2 Genetics, Heredity, and Molecular Biology 7.3 Human Body Systems, Ecology, and Evolution 7.4 Matter, Phase Changes, and Atomic Structure 7.5 Periodic Trends, Chemical Bonding, Reactions, Acids and Bases
8 Chapter 8: General Science II — Physics and Earth Science
8.1 Mechanics: Motion, Newton's Laws, Work, and Energy 8.2 Waves, Sound, Light, Heat, Electricity, and Magnetism 8.3 Earth's Structure, Plate Tectonics, and the Rock Cycle 8.4 The Atmosphere, Weather, the Water Cycle, and Astronomy
9 Chapter 9: Aptitude and Abstract Reasoning
9.1 Figure Series and Pattern Progression 9.2 Matrix Reasoning and Figural Analogies 9.3 Spatial Reasoning: Rotation, Reflection, and Chirality 9.4 Paper Folding, Cube Nets, and Orthographic Views 9.5 Deductive Reasoning, Syllogisms, and Venn Diagrams 9.6 Number, Letter, and Alphanumeric Series 9.7 Odd-One-Out, Classification, and Logic Puzzles
2.4 Radicals, Rational Expressions, and Scientific Notation
Key Takeaways Explain This Topic AI
Simplify a radical by extracting the largest perfect-square factor: 180 = 36 ⋅ 5 = 6 5 \sqrt{180} = \sqrt{36 \cdot 5} = 6\sqrt{5} 180 = 36 ⋅ 5 = 6 5 , and for cube roots 54 3 = 3 2 3 \sqrt[3]{54} = 3\sqrt[3]{2} 3 54 = 3 3 2 .
Radicals add only when the radicands match, so 50 + 3 18 − 8 = 12 2 \sqrt{50} + 3\sqrt{18} - \sqrt{8} = 12\sqrt{2} 50 + 3 18 − 8 = 12 2 , but 9 + 16 = 5 \sqrt{9 + 16} = 5 9 + 16 = 5 and never 3 + 4 3 + 4 3 + 4 .
Rationalise a two-term denominator with its conjugate: 6 5 − 2 \frac{6}{\sqrt{5} - \sqrt{2}} 5 − 2 6 becomes 2 ( 5 + 2 ) 2(\sqrt{5} + \sqrt{2}) 2 ( 5 + 2 ) because ( 5 − 2 ) ( 5 + 2 ) = 3 (\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2}) = 3 ( 5 − 2 ) ( 5 + 2 ) = 3 .
Squaring both sides can create extraneous roots: x + 7 = x − 5 \sqrt{x + 7} = x - 5 x + 7 = x − 5 yields x = 9 x = 9 x = 9 and x = 2 x = 2 x = 2 , but only x = 9 x = 9 x = 9 satisfies the original equation.
Excluded values come from the original denominators, so x 2 − 9 x 2 + x − 12 \frac{x^2-9}{x^2+x-12} x 2 + x − 12 x 2 − 9 simplifies to x + 3 x + 4 \frac{x+3}{x+4} x + 4 x + 3 yet still bars both x = 3 x = 3 x = 3 and x = − 4 x = -4 x = − 4 .
Last updated: August 2026
2.4 Radicals, Rational Expressions, and Scientific Notation
Radicals and rational expressions are where careless algebra becomes visibly wrong: a root that does not satisfy the original equation, a "cancellation" that was never legal, a coefficient outside the range that scientific notation permits. Every number used below is chosen to come out clean, so the whole section can be worked with pen and paper. The laws of exponents and logarithms belong to a neighbouring section; here they are used, not re-taught.
1. Simplifying Radicals
A radical is simplified when no perfect-square factor (or perfect-cube factor, for cube roots) remains under the sign.
Rule Statement Example
Product a ⋅ b = a b \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} a ⋅ b = ab 3 ⋅ 12 = 36 = 6 \sqrt{3} \cdot \sqrt{12} = \sqrt{36} = 6 3 ⋅ 12 = 36 = 6
Quotient a b = a b \sqrt{\dfrac{a}{b}} = \dfrac{\sqrt{a}}{\sqrt{b}} b a = b a 49 16 = 7 4 \sqrt{\dfrac{49}{16}} = \dfrac{7}{4} 16 49 = 4 7
Extraction a 2 b = a b \sqrt{a^2 b} = a\sqrt{b} a 2 b = a b 180 = 36 ⋅ 5 = 6 5 \sqrt{180} = \sqrt{36 \cdot 5} = 6\sqrt{5} 180 = 36 ⋅ 5 = 6 5
Cube roots a 3 b 3 = a b 3 \sqrt[3]{a^3 b} = a\sqrt[3]{b} 3 a 3 b = a 3 b 54 3 = 27 ⋅ 2 3 = 3 2 3 \sqrt[3]{54} = \sqrt[3]{27 \cdot 2} = 3\sqrt[3]{2} 3 54 = 3 27 ⋅ 2 = 3 3 2
Fractional exponent a m / n = a m n = ( a n ) m a^{m/n} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m a m / n = n a m = ( n a ) m 16 3 / 4 = ( 16 4 ) 3 = 2 3 = 8 16^{3/4} = \left(\sqrt[4]{16}\right)^3 = 2^3 = 8 1 6 3/4 = ( 4 16 ) 3 = 2 3 = 8
Not a rule a + b ≠ a + b \sqrt{a + b} \neq \sqrt{a} + \sqrt{b} a + b = a + b 9 + 16 = 5 \sqrt{9+16} = 5 9 + 16 = 5 , but 3 + 4 = 7 3 + 4 = 7 3 + 4 = 7
Odd roots accept negative radicands ( − 125 3 = − 5 \sqrt[3]{-125} = -5 3 − 125 = − 5 ) while even roots of negatives are not real numbers.
Radicals combine by addition only when the radicands match, exactly as 5 k + 9 k − 2 k = 12 k 5k + 9k - 2k = 12k 5 k + 9 k − 2 k = 12 k : 50 + 3 18 − 8 = 5 2 + 9 2 − 2 2 = 12 2 \sqrt{50} + 3\sqrt{18} - \sqrt{8} = 5\sqrt{2} + 9\sqrt{2} - 2\sqrt{2} = 12\sqrt{2} 50 + 3 18 − 8 = 5 2 + 9 2 − 2 2 = 12 2
2. Rationalising Denominators
Answer keys are written without radicals in the denominator, so learn both moves.
Monomial denominator — multiply top and bottom by that radical: 7 21 = 7 21 21 = 21 3 \dfrac{7}{\sqrt{21}} = \dfrac{7\sqrt{21}}{21} = \dfrac{\sqrt{21}}{3} 21 7 = 21 7 21 = 3 21 .
Two-term denominator — multiply by the conjugate , the same two terms with the middle sign reversed, because ( a − b ) ( a + b ) = a 2 − b 2 (a-b)(a+b) = a^2 - b^2 ( a − b ) ( a + b ) = a 2 − b 2 destroys both radicals: 6 5 − 2 ⋅ 5 + 2 5 + 2 = 6 ( 5 + 2 ) 5 − 2 = 2 ( 5 + 2 ) \frac{6}{\sqrt{5} - \sqrt{2}} \cdot \frac{\sqrt{5} + \sqrt{2}}{\sqrt{5} + \sqrt{2}} = \frac{6(\sqrt{5} + \sqrt{2})}{5 - 2} = 2\left(\sqrt{5} + \sqrt{2}\right) 5 − 2 6 ⋅ 5 + 2 5 + 2 = 5 − 2 6 ( 5 + 2 ) = 2 ( 5 + 2 )
Fractional exponents make some values easier than the radical form suggests: 27 − 2 / 3 = 1 ( 27 3 ) 2 = 1 9 27^{-2/3} = \dfrac{1}{\left(\sqrt[3]{27}\right)^2} = \dfrac{1}{9} 2 7 − 2/3 = ( 3 27 ) 2 1 = 9 1 . A negative exponent inverts the base; it never makes the value negative.
- 3. Radical Equations and Extraneous Roots
- Worked Example 1
Solve x + 7 = x − 5 \sqrt{x + 7} = x - 5 x + 7 = x − 5 .
Square both sides: x + 7 = x 2 − 10 x + 25 x + 7 = x^2 - 10x + 25 x + 7 = x 2 − 10 x + 25 .
Rearrange: x 2 − 11 x + 18 = 0 x^2 - 11x + 18 = 0 x 2 − 11 x + 18 = 0 .
Factor: ( x − 9 ) ( x − 2 ) = 0 (x - 9)(x - 2) = 0 ( x − 9 ) ( x − 2 ) = 0 , so x = 9 x = 9 x = 9 or x = 2 x = 2 x = 2 .
Check both in the original equation. For x = 9 x = 9 x = 9 : 16 = 4 \sqrt{16} = 4 16 = 4 and 9 − 5 = 4 9 - 5 = 4 9 − 5 = 4 , so it holds. For x = 2 x = 2 x = 2 : 9 = 3 \sqrt{9} = 3 9 = 3 but 2 − 5 = − 3 2 - 5 = -3 2 − 5 = − 3 , so it fails.
The only solution is x = 9 x = 9 x = 9 ; x = 2 x = 2 x = 2 is extraneous . Squaring is not a reversible step — it erases sign information, so it can manufacture roots that were never solutions. Checking is part of the method, not optional neatness.
4. Rational Expressions
A rational expression is a ratio of polynomials. Simplify by factoring completely and cancelling common factors — never individual terms. x 2 − 9 x 2 + x − 12 = ( x − 3 ) ( x + 3 ) ( x + 4 ) ( x − 3 ) = x + 3 x + 4 \frac{x^2 - 9}{x^2 + x - 12} = \frac{(x-3)(x+3)}{(x+4)(x-3)} = \frac{x+3}{x+4} x 2 + x − 12 x 2 − 9 = ( x + 4 ) ( x − 3 ) ( x − 3 ) ( x + 3 ) = x + 4 x + 3
The excluded values come from the original denominator, so both x = 3 x = 3 x = 3 and x = − 4 x = -4 x = − 4 remain barred even though ( x − 3 ) (x-3) ( x − 3 ) disappeared.
Operations follow ordinary fraction rules:
Multiply: factor, cross-cancel, then multiply.
Divide: multiply by the reciprocal of the second expression.
Add or subtract: build the least common denominator from the factored denominators.
Worked Example 2 — addition and a complex fraction
Combine 3 x − 2 + 5 x + 1 \dfrac{3}{x-2} + \dfrac{5}{x+1} x − 2 3 + x + 1 5 . The LCD is ( x − 2 ) ( x + 1 ) (x-2)(x+1) ( x − 2 ) ( x + 1 ) , so the sum is 3 ( x + 1 ) + 5 ( x − 2 ) ( x − 2 ) ( x + 1 ) = 3 x + 3 + 5 x − 10 ( x − 2 ) ( x + 1 ) = 8 x − 7 ( x − 2 ) ( x + 1 ) \frac{3(x+1) + 5(x-2)}{(x-2)(x+1)} = \frac{3x + 3 + 5x - 10}{(x-2)(x+1)} = \frac{8x - 7}{(x-2)(x+1)} ( x − 2 ) ( x + 1 ) 3 ( x + 1 ) + 5 ( x − 2 ) = ( x − 2 ) ( x + 1 ) 3 x + 3 + 5 x − 10 = ( x − 2 ) ( x + 1 ) 8 x − 7
Now simplify the complex fraction 1 x − 1 3 x − 3 \dfrac{\frac{1}{x} - \frac{1}{3}}{x - 3} x − 3 x 1 − 3 1 . The numerator becomes 3 − x 3 x \dfrac{3 - x}{3x} 3 x 3 − x , and dividing by ( x − 3 ) (x-3) ( x − 3 ) gives 3 − x 3 x ( x − 3 ) = − ( x − 3 ) 3 x ( x − 3 ) = − 1 3 x \frac{3-x}{3x(x-3)} = \frac{-(x-3)}{3x(x-3)} = -\frac{1}{3x} 3 x ( x − 3 ) 3 − x = 3 x ( x − 3 ) − ( x − 3 ) = − 3 x 1
Recognising that 3 − x = − ( x − 3 ) 3 - x = -(x - 3) 3 − x = − ( x − 3 ) is the whole trick, and it appears constantly.
5. Rational Equations
Clear denominators by multiplying every term by the LCD, then solve — but test each answer against the excluded values.
Solve 5 x − 1 2 = 3 2 x \dfrac{5}{x} - \dfrac{1}{2} = \dfrac{3}{2x} x 5 − 2 1 = 2 x 3 . Multiplying through by 2 x 2x 2 x gives 10 − x = 3 10 - x = 3 10 − x = 3 , so x = 7 x = 7 x = 7 . Since 7 is not excluded, it is valid; substituting confirms 5 7 − 1 2 = 10 − 7 14 = 3 14 \frac{5}{7} - \frac{1}{2} = \frac{10-7}{14} = \frac{3}{14} 7 5 − 2 1 = 14 10 − 7 = 14 3 .
Contrast x x − 4 = 4 x − 4 + 3 \dfrac{x}{x-4} = \dfrac{4}{x-4} + 3 x − 4 x = x − 4 4 + 3 . Multiplying by ( x − 4 ) (x-4) ( x − 4 ) gives x = 4 + 3 ( x − 4 ) = 3 x − 8 x = 4 + 3(x-4) = 3x - 8 x = 4 + 3 ( x − 4 ) = 3 x − 8 , so x = 4 x = 4 x = 4 — but x = 4 x = 4 x = 4 makes the original denominators zero. The equation has no solution , and "no solution" is a genuine answer option.
6. Scientific Notation
Scientific notation writes a number as a × 10 n a \times 10^n a × 1 0 n where 1 ≤ ∣ a ∣ < 10 1 \le \lvert a \rvert < 10 1 ≤ ∣ a ∣ < 10 and n n n is an integer. Move the decimal point until exactly one non-zero digit stands to its left; the number of places moved is ∣ n ∣ \lvert n \rvert ∣ n ∣ , and n n n is negative for numbers below 1.
- Standard form Scientific notation
- 4,780,000 4{,}780{,}000 4 , 780 , 000 4.78 × 10 6 4.78 \times 10^6 4.78 × 1 0 6
- 93,000 93{,}000 93 , 000 9.3 × 10 4 9.3 \times 10^4 9.3 × 1 0 4
- 570 570 570 km in metres 5.7 × 10 5 5.7 \times 10^5 5.7 × 1 0 5
- 0.058 0.058 0.058 5.8 × 10 − 2 5.8 \times 10^{-2} 5.8 × 1 0 − 2
- 0.00000342 0.00000342 0.00000342 3.42 × 10 − 6 3.42 \times 10^{-6} 3.42 × 1 0 − 6
Multiplying: multiply the coefficients, add the exponents. ( 3.0 × 10 8 ) ( 4.5 × 10 − 3 ) = 13.5 × 10 5 (3.0 \times 10^8)(4.5 \times 10^{-3}) = 13.5 \times 10^5 ( 3.0 × 1 0 8 ) ( 4.5 × 1 0 − 3 ) = 13.5 × 1 0 5 , which must be renormalised to 1.35 × 10 6 1.35 \times 10^6 1.35 × 1 0 6 because 13.5 is not below 10.
Dividing: divide the coefficients, subtract the exponents. 8.4 × 10 − 2 2.1 × 10 5 = 4.0 × 10 − 7 \dfrac{8.4 \times 10^{-2}}{2.1 \times 10^{5}} = 4.0 \times 10^{-7} 2.1 × 1 0 5 8.4 × 1 0 − 2 = 4.0 × 1 0 − 7 .
Comparing: the larger exponent wins outright. 6.0 × 10 − 4 6.0 \times 10^{-4} 6.0 × 1 0 − 4 beats 9.0 × 10 − 5 9.0 \times 10^{-5} 9.0 × 1 0 − 5 despite the smaller coefficient, and the ratio 6.0 9.0 × 10 1 ≈ 6.7 \frac{6.0}{9.0} \times 10^{1} \approx 6.7 9.0 6.0 × 1 0 1 ≈ 6.7 says it is nearly seven times as large.
Common Traps
Splitting a radical across addition: x 2 + 9 \sqrt{x^2 + 9} x 2 + 9 is not x + 3 x + 3 x + 3 .
Skipping the check after squaring, and keeping an extraneous root.
Cancelling terms instead of factors: x + 3 3 \dfrac{x+3}{3} 3 x + 3 does not reduce to x x x .
Reporting excluded values from the simplified expression only.
Leaving an answer as 13.5 × 10 5 13.5 \times 10^5 13.5 × 1 0 5 or 0.42 × 10 − 3 0.42 \times 10^{-3} 0.42 × 1 0 − 3 ; neither coefficient is in [ 1 , 10 ) [1, 10) [ 1 , 10 ) .
Assuming 10 − 6 10^{-6} 1 0 − 6 makes a quantity negative — it makes it small.