The decision and its conditions
Read a rational exponent as a root followed by a power. An odd root of a negative number is negative, unlike the nonnegative principal square root. Scientific notation separates a coefficient satisfying 1≤c<10 from a power of ten. After multiplying coefficients you may need to normalize again, changing the exponent to keep the value unchanged. A large exponent indicates scale, not a number of significant digits. Compare magnitudes before using a calculator and distinguish a ratio, which answers how many times, from a difference, which retains the original unit.
Worked example
What real number cubed equals -27? An odd power preserves a negative sign; -27 is negative. 3³=27, so (−3)³=-27. The required real number is -3.
A contrasting decision
Two geometrically similar solids have volume ratio small:large = 1:27. What is the larger solid’s surface area divided by the smaller solid’s surface area? For similar solids, volume scales as the cube of the length factor, so the length factor is ∛27=3. Surface area uses two length dimensions, so square that length factor. The large-to-small surface-area factor is 3²=9; it is not the volume factor 27.
Diagnose a competing method
The first example’s tempting answer 3 follows this mistake: Apply the nonnegative square-root convention to an odd root. Unlike a square, a cube can be negative. Find the magnitude first.
Transfer before revealing
The product (2 × 10^4)(3 × 10^3) is written 6 × 10^k. What is k? Pause and identify the requested quantity. Solution: Multiply coefficients separately from powers of ten. 2×3=6 and equal-base exponents add: 4+3=7. The coefficient 6 already satisfies 1≤6<10, so k=7.
Check the boundary of the method
A large exponent indicates scale, not a number of significant digits. Recover the length scale from the volume ratio, then use two dimensions to obtain the area scale. Compare the answer with the question’s units and conditions. One correct example does not prove an identity or a statistical generalization.
What real number cubed equals -27?
- An odd power preserves a negative sign; -27 is negative.
- 3³=27, so (−3)³=-27.
- The required real number is -3.
Compare the other answer choices
Apply the nonnegative square-root convention to an odd root.
Return the square of the positive cube root.
Divide by three instead of taking a cube root.
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.
GED Mathematical Reasoning Assessment Guide for Educators