Goal and prerequisites
Simplify a surd, rationalise a denominator and interpret a fractional power without rounding. Review number structure before starting.
Method and its conditions
Extract square factors before combining like surds. Rationalise by multiplying numerator and denominator by the same nonzero expression. Read a positive-base fractional exponent m/n as taking the nth root and then the mth power.
Worked example: show the decisions
√72+√18=6√2+3√2=9√2. For 5/√3, multiply by √3/√3 to obtain 5√3/3. A different rationalisation uses conjugates: 1/(√5+2)=(√5−2)/(5−4)=√5−2. Each step preserves the exact value.
Contrast: diagnose an incorrect approach
√a+√b is not generally √(a+b). Likewise, rationalising is not permission to change just the denominator. Check a conjugate product using difference of squares, and remember that the exponent 2/3 describes a root and a power, not multiplication by two thirds.
Transfer: try a changed situation
Simplify √48−√12 and evaluate 64^(2/3). Explain why neither calculation needs a decimal approximation.
Transfer solution and interpretation
The surds become 4√3−2√3=2√3. The cube root of 64 is 4, so 64^(2/3)=4²=16. Square factors and exact integer roots preserve exactness.
Combine roots, negative powers and division
For 64^(−2/3) ÷ 4^(−1), the cube root is 4. The first power is 4^(−2)=1/16 and the divisor is 1/4. Dividing gives (1/16)×4=1/4. Multiplying the two reciprocals instead would answer a different operation.
Simplify √72+√162 as c√2. Find c.
- Extract square factors: √72=√(36×2)=6√2 and √162=√(81×2)=9√2.
- Add the coefficients of the same √2: 6+9=15.
- The expression is 15√2, so c=15.
Compare the other answer choices
Subtract the two coefficients instead of adding them.
Multiply the two surd coefficients instead of adding them.
Add the radicands and report that sum as a coefficient.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
0580 syllabus for exams in 2025–2027