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bhanu29
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12 Feb 2026, 06:33
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D
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
35%
(medium)
Question Stats:
71% (02:14) correct
29%
(01:57)
wrong
based on 91
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History
Date
Time
Result
Not Attempted Yet
If \((\frac{1}{5})^a * (\frac{1}{20})^b =\frac{1}{125*100^{10}}\), \(a+b = ?\)
A. 13
B. 17
C. 20
D. 23
E. 26
A. 13
B. 17
C. 20
D. 23
E. 26
12 Feb 2026, 12:57
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This exponents problem requires breaking everything down to prime bases. The key concept being tested is your ability to manipulate exponents with the same base.
**Step 1: Convert everything to prime bases (5 and 2)**
Left side:
- (1/5)^a = 5^(-a)
- (1/20)^b = (1/(4×5))^b = (1/(22 × 5))^b = 2^(-2b) × 5^(-b)
So the left side becomes: 5^(-a) × 2^(-2b) × 5^(-b) = 2^(-2b) × 5^(-a-b)
Right side:
- 125 = 53
- 100 = 102 = (2×5)2 = 22 × 52
- 100^10 = 2^20 × 5^20
So: 1/(125 × 100^10) = 1/(53 × 2^20 × 5^20) = 2^(-20) × 5^(-23)
**Step 2: Set up equations by matching exponents**
Now we have: 2^(-2b) × 5^(-a-b) = 2^(-20) × 5^(-23)
For this equality to hold, the exponents of each base must be equal:
For base 2: -2b = -20, so **b = 10**
For base 5: -a - b = -23, so **a + b = 23**
**Answer: (D) 23**
**Alternative approach (if you saw b=10):**
Once you know b = 10, you could solve: -a - 10 = -23, which gives a = 13.
Therefore a + b = 13 + 10 = 23.
**Common trap:** Students often make arithmetic errors when dealing with negative exponents or forget to fully break down 100^10 into prime factors. Taking it slow and being systematic with prime factorization prevents mistakes.
**Takeaway:** For exponent equations, convert everything to the same prime bases and match exponents. This technique works every time and is especially useful in GMAT Focus, where exponent manipulation appears frequently.
**Step 1: Convert everything to prime bases (5 and 2)**
Left side:
- (1/5)^a = 5^(-a)
- (1/20)^b = (1/(4×5))^b = (1/(22 × 5))^b = 2^(-2b) × 5^(-b)
So the left side becomes: 5^(-a) × 2^(-2b) × 5^(-b) = 2^(-2b) × 5^(-a-b)
Right side:
- 125 = 53
- 100 = 102 = (2×5)2 = 22 × 52
- 100^10 = 2^20 × 5^20
So: 1/(125 × 100^10) = 1/(53 × 2^20 × 5^20) = 2^(-20) × 5^(-23)
**Step 2: Set up equations by matching exponents**
Now we have: 2^(-2b) × 5^(-a-b) = 2^(-20) × 5^(-23)
For this equality to hold, the exponents of each base must be equal:
For base 2: -2b = -20, so **b = 10**
For base 5: -a - b = -23, so **a + b = 23**
**Answer: (D) 23**
**Alternative approach (if you saw b=10):**
Once you know b = 10, you could solve: -a - 10 = -23, which gives a = 13.
Therefore a + b = 13 + 10 = 23.
**Common trap:** Students often make arithmetic errors when dealing with negative exponents or forget to fully break down 100^10 into prime factors. Taking it slow and being systematic with prime factorization prevents mistakes.
**Takeaway:** For exponent equations, convert everything to the same prime bases and match exponents. This technique works every time and is especially useful in GMAT Focus, where exponent manipulation appears frequently.
