Powers Check Practice Test Worksheets
Powers check questions ask which expression does not have the same value as the original. Writing each expression using prime factors can help you compare them without multiplying out large numbers.
For students who know multiplication facts, but need practice recognising powers, squares, and repeated factors quickly.
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What Do Powers Check Questions Look Like?
Powers check questions test your ability to recognise equivalent algebraic expressions by breaking numbers down into prime factors and applying exponent rules. The goal is to spot the one expression that does not match the original value. Here is a worked example.
Spotting One Extra Power
Question: 93 × 8 is not equal to which option?
A. 23 × 37
B. 22 × 3 × 32 × 32 × 2 × 3
C. 6 × 12 × 34
D. 4 × 93 × 2
E. 36 × 23
Worked Method
Step 1: 93 = (32)3 = 36, and 8 = 23.
Step 2: The original expression is 23 × 36.
Step 3: Option A is 23 × 37, which has one extra factor of 3.
Step 4: Check that all four remaining options match the original:
- B: 22+1 × 31+2+2+1 = 23 × 36.
- C: (2 × 3) × (22 × 3) × 34 = 23 × 36.
- D: 22 × 36 × 2 = 23 × 36.
- E: 36 × 23 = 23 × 36; changing the order of factors does not change the value.
Answer: A. Tip: Compare exponents, not how complicated the option looks.
Rules to Help You Solve Powers Check Questions
To solve powers check questions like the example above, simplify each expression into its prime-power form. A power is a short way to write repeated multiplication (like 34 = 3 × 3 × 3 × 3), and comparing prime bases avoids multiplying out large numbers. Keep these key exponent rules in mind:
🔎 Rule 1: Convert to prime factors
Rewrite numbers such as 4, 8, and 9 using prime factors: 4 = 22, 8 = 23, and 9 = 32.
Example: 93 × 8 = (32)3 × 23 = 23 × 36.
Different-looking expressions can have the same value. Compare their prime factors rather than their appearance.
Matching example: 6 × 12 × 34 = (2 × 3) × (22 × 3) × 34 = 21+2 × 31+1+4 = 23 × 36.
➕ Rule 2: Multiplying powers with the same base
When powers with the same base are multiplied, add their exponents.
Example: 33 × 32 × 3 = 33+2+1 = 36.
🔁 Rule 3: Raising a power to another power
When a power is raised to another power, multiply the exponents. The brackets show which power is repeated.
Example: (32)3 = 32 × 32 × 32 = 32×3 = 36.
🧱 Rule 4: Keep each prime base separate
Once each expression is in prime-power form, compare the exponent of each prime, including any 5s, 7s, or other primes. If a prime is missing or an exponent is different, the values do not match.
Example: 23 × 36 is not the same as 23 × 37.
Two More Worked Examples
Here are two more examples showing how to factorise composite numbers and avoid missing hidden prime factors.
Combining Several Factors
Question: 3 × 8 × 9 × 9 is not equal to which option?
A. 34 × 3 × 22 × 2
B. 4 × 92 × 2 × 3
C. 6 × 12 × 33
D. 23 × 34
E. 34 × 3 × 23
Worked Method
Step 1: Convert the composite numbers: 8 = 23 and each 9 = 32.
Step 2: Combine the 3s: 3 × 32 × 32 = 31+2+2 = 35.
Step 3: The original is 23 × 35. Option D is 23 × 34, so it is missing one factor of 3.
Step 4: Check that all four remaining options match the original:
- A: 22+1 × 34+1 = 23 × 35.
- B: 22 × 34 × 2 × 3 = 23 × 35.
- C: (2 × 3) × (22 × 3) × 33 = 23 × 35.
- E: 34+1 × 23 = 23 × 35.
Answer: D. Tip: A plain 3 counts as 31.
Similar-Looking Bases Can Hide a Missing Factor
Question: 22 × 9 × 42 is not equal to which option?
A. 43 × 9
B. 26 × 3
C. 32 × 24 × 22
D. 22 × 2 × 23 × 32
E. 22 × 23 × 2 × 3 × 3
Worked Method
Step 1: 9 = 32 and 42 = (22)2 = 24.
Step 2: Combine the 2s: 22 × 24 = 26, so the original is 26 × 32.
Step 3: Option B is 26 × 3. It has the correct power of 2 but only 31, not 32.
Step 4: Check that all four remaining options match the original:
- A: (22)3 × 32 = 26 × 32.
- C: 32 × 24+2 = 26 × 32.
- D: 22+1+3 × 32 = 26 × 32.
- E: 22+3+1 × 31+1 = 26 × 32.
Answer: B. Tip: Do not stop after checking the power of 2. The power of 3 matters too.
