Mathematics · 02 / 23
Powers and roots: count the factors
Middle school · Worked example and 4 exercises with solutions, at your pace.
The principle, simply
A positive integer power counts equal factors. For nonzero a, a⁰ = 1 and a⁻ⁿ = 1/aⁿ. The real square root of a ≥ 0 is the nonnegative number whose square is a. Hence √(x²) = |x|: the principal square root cannot be negative.
- 01Equal factors
- 02Matching rule
- 03Root domain
- 04Check the sign
WORKED EXAMPLE
From principle to reasoning
2³ × 2⁴ = 2⁷ = 128 joins seven factors. (2³)⁴ = 2¹² uses four groups of three. √((-3)²) = √9 = 3.
Solve a problem in this subject
- Define the objects
State unknowns, domain and data. Real square roots need nonnegative arguments; denominators must be nonzero.
- Choose a transformation
Name the rule: expansion, equivalent equations, factoring, differentiation. Check its conditions. Squaring an equation requires testing candidates in the original.
- Show the reasoning
Each line should follow from the previous one. Keep exact values where possible and distinguish equality, approximation and implication.
- Test and conclude
Substitute into the original expression. Check graph points and domain boundaries. State a solution set or the required unit.
YOUR TURN
01 · Try before checking
Simplify 5³/5² and find √((-4)²).
One hint
Cancel common factors; the principal square root is nonnegative.
Reveal and understand the solution
5³/5² = 5. √((-4)²) = 4, not −4.
Compare method, conditions and answer; identify the exact part to revisit.
IMPROVE THROUGH PRACTICE
Three additional challenges with solutions
Start with the first challenge. Write your method before opening a hint; use the solution to find the first step that needs work.
Simplify 2³ × 2⁴. Why not multiply the exponents?
A hint to get started
Count the factors of 2.
Step-by-step solution
- There are three factors followed by four more.
- Together they make 2⁷ = 128.
- Exponents multiply for a power of a power, such as (2³)⁴.
Which step did you miss? Retry without help, then explain why the method works.
Simplify √72 as a√b with square-free b.
A hint to get started
Find a large square factor of 72.
Step-by-step solution
- 72 = 36 × 2.
- √72 = √36 × √2 = 6√2.
- A square root does not distribute over addition.
Which step did you miss? Retry without help, then explain why the method works.
For x = −3, compare √(x²) and x.
A hint to get started
A square root denotes the nonnegative result.
Step-by-step solution
- x² = 9, so √9 = 3.
- The result is
- x
- , here 3, not −3.
- Thus √(x²) = x requires x ≥ 0.
Which step did you miss? Retry without help, then explain why the method works.
The trap to avoid
In general √(a + b) is not √a + √b. −2² = −4 whereas (−2)² = 4.
Without notes, can you…
- Explain the principle in your own words?
- Rebuild the diagram and its connections?
- Solve the exercise and justify the method?
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