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12 Days of Christmas GMAT Competition 2025 - Day 2: If (3.7 * 10^a)

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Nitish03

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Use a^m * a^n = a^ m+n
So,10^(a+b-2)=10^(c-4)(....37*58=2146 hence will cancel out)
As the base is equal,
c-4= a+b-2
c= a+b+2
Hence, option B.

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B. a+b+2

3.7 x 5.8 X 10^a x 10^b = 0.2146 x 10^c
21.46 x 10^(a+b) = 0.2146 x 10^c
10^(a+b)= 10^c / 100
10^(a+b) x 10^2 = 10^c
10^(a+b+2) = 10^c
c= a+b+2

voluptatumsint

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(3.7*10^a) * (5.8*10^b) = 0.2146

Solving for C in terms of a and b.

Can be initally somewhat confuising, but using my human brain I reread the prompt, and understand I am being asked what is the value of C.

(3.7*10^a) * (5.8*10^b) is equivilant to

10^a*10^b*5.8*3.7 or

By rules of exponets I can condense this into

10^(a+b)*5.8*3.7 | 10^(a+b)*21.46 = 0.2146 * 10^C

I recognize that 21.46 is 100 ( or 10^2) times more than .2146 | There are a few different steps that can be taken mentally or on paper here to make the equations equal and come to the answer.

One of the would be to move the decimal of 21.46 over two places and multiply my 10^2 | 10^2*10^(a+b)*.2146 = .2146*10^c | In which case, we condense terms and come to the conclusion that a+b+2 = c

Or

We can make the right side of the equation look like the left. or multiply 10^2 out of c.

10^(a+b)*21.46 = 21.46 * 10^(c-2)

Divide by 21.46

10^(a+b)=10^(c-2)

Drop bases as they are the same, raised to the same power

a+b = c- 2

a+b+2 = c

Bunuel

If $(3.7 * 10^a) * (5.8 * 10^b) = 0.2146 * 10^c$, what is c in terms of a and b?

A. a + b - 2
B. a + b + 2
C. (a + b)/2
D. ab - 2
E. ab + 2

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Answer: B

Bunuel

If $(3.7 * 10^a) * (5.8 * 10^b) = 0.2146 * 10^c$, what is c in terms of a and b?

A. a + b - 2
B. a + b + 2
C. (a + b)/2
D. ab - 2
E. ab + 2

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When you multiply the terms you are left with 21.46 x 10^a+b. Exponent rules tell us that when we multiply powers of 10 we can add the exponents. If we rewrite the right side of the equation to match the left, we must move the decimal two to the left, meaning we add two to the exponent. With the coefficients now the same, we can look at the exponents following or previous steps and see that a + b + 2 = c

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16 Dec 2025, 14:19

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(3.7*10^a) * (5.8 *10^b) =2.146*10^-1*10^c
21.465*10^a*10^b*10^2 =2.146*10^-1*10^c
2.0146*10^1*10^a+b= 2.1465*10^c-1
a+b+1= c-1
c= a+b+2

Bunuel

If $(3.7 * 10^a) * (5.8 * 10^b) = 0.2146 * 10^c$, what is c in terms of a and b?

A. a + b - 2
B. a + b + 2
C. (a + b)/2
D. ab - 2
E. ab + 2

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KunchiGoks

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Conceptual refresh:
1. Product rule of exponents: when exponents with the same base are multiplied, we can add the exponents 10^a x 10^b = 10^a+b.
2. When multiplying with powers of 10, shifting the decimal places requires adjusting the exponent by the same number of places i.e. shifting to the right decreases the exponent and shifting to the left increases the exponent.

a) Multiply the decimals 3.7 x 5.8 = 21.46
b) To match this with the other side of the equation from 0.2146 to 21.46 would require shifting 2 decimal places to the right hence the exponent C would decrease by two decimal places 10^C - 2.
c) Once the decimal parts across both sides of the equation match, we can equate the exponents since:
a + b = c - 2 therefore:
a + b + 2 = c

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16 Dec 2025, 17:47

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Find solution in attached image

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WhatsApp Image 2025-12-17 at 06.16.16.jpeg [ 39.93 KiB | Viewed 238 times ]

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Steps :
1. Multiply the numbers
2. Combine the powers

left side will become 3.7*5.8*10^(a+b) = 21.46 * 10^(a+b)

rewrite to match the right side 21.46 * 10^-2 *10^c

Comparing
10^(a+b) to 10^(c-2)

a+b= c-2
hence c= a+b+2

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16 Dec 2025, 19:11

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(3.7*10^a)*(5.8*10^b) = 0.2146 * 10^c
21.46 * 10^a * 10^b = 0.2146 * 10^c
0.2146 * 10^2 * 10^a * 10^b = 0.2146 * 10^c

same base, we can add the exponents:

0.2146 * 10^(a+b+2) = 0.2146 * 10^c

c = a+b+2

IMO B

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(3.7x10^a) x(5.8x10^b) = 0.2146 x 10^c
[(37/10)x10^a] x[(58/10)x10^b) = (2146/10000)x10^c
(2146/100)x10^(a+b) = 2146x10^(c-4)
Now cancelling 2146 on both sides
10^(a+b-2) = 10^(c-4)

So ,

a+b-2=c-4
c = a+b+2

Bunuel

If $(3.7 * 10^a) * (5.8 * 10^b) = 0.2146 * 10^c$, what is c in terms of a and b?

A. a + b - 2
B. a + b + 2
C. (a + b)/2
D. ab - 2
E. ab + 2

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16 Dec 2025, 19:27

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if 3.7 810^a * 5.8 *10^b= 0.2146 * 10^c
now we can make it terms of 3.7 and 5.8 can be written as
3.7 *5.5 * 10^C+2

so a*b=c+2
c can be written as a*b-2=C

hence option D is correct

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16 Dec 2025, 19:29

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Playing with mathematical expression:

(3.7*10^a) * (5.8*10^b) = 21.46 * 10^a * 10^b = 0.2146 * 10^2 * 10^a * 10^b = 0.2146 * 10^(a+b+2) = 0.2146 * 10^c

The exponents of base 10 must be the same:

c=a+b+2

Answer B

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16 Dec 2025, 19:47

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Bunuel

If $(3.7 * 10^a) * (5.8 * 10^b) = 0.2146 * 10^c$, what is c in terms of a and b?

A. a + b - 2
B. a + b + 2
C. (a + b)/2
D. ab - 2
E. ab + 2

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Clearly 3.7 * 5.8 is 21.46. As you need to move two spots to the left to get 0.2146, this means 21.46 / 10^2. Then a+b-2=c

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3.7 * 10^a * 5.8 * 10^b = 3.7*5.8 * 10^a * 10^b = 21.46 * 10^(a+b)

0.2146 * 10^c = 21.46 * 10^(-2) * 10^c = 21.46 * 10^(c-2)

Equating:
a+b=c-2
c=a+b+2

The answer is B

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Bunuel

If $(3.7 * 10^a) * (5.8 * 10^b) = 0.2146 * 10^c$, what is c in terms of a and b?

A. a + b - 2
B. a + b + 2
C. (a + b)/2
D. ab - 2
E. ab + 2

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$(3.7 * 10^a) * (5.8 * 10^b) = 0.2146 * 10^c$

$(3.7 * 10^a) * (5.8 * 10^b) = 21.56 * 10 ^ (a+b) $
$ = 0.2156 * 10^2 * 10^(a+b) $
$ = 0,2156 * 10^(a+b+2) $

Therefore c= a+b+2
Correct Answer: B

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On computing 3.7 * 5.8, we get 21.46. So the entire computation becomes 21.46 * 10^(a + b).
Now on RHS, we have 0.2146 * 10^(c).
Equating both the sides, we get
21.46 * 10^(a+b) = 0.2146 * 10^c
=> 21.46 * 10^(a+b) = 21.46 * 10^(c-2)
Equating the bases, now the powers are equal, so a + b = c - 2 => c = a + b + 2.
The answer is B.

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Extending the first term:

(3.7*10^a)*(5.8*10^b) = 21.46*10^(a+b) = 0.2146*10^(a+b+2)

Adding the second term:

0.2146*10^(a+b+2) = 0.2146*10^c

c=a+b+2

The correct answer is B

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remember that $x^m * x^n = x^{m+n}$

$37 * 58 * 10^{-2} * 10^{a+b} = 2146 * 10^{-4+c}$

simplifying and equating powers of 10
$-2+a+b = -4+c$

$a+b+2 = c$

ans: option B

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to find c in tems of a and b we need to simplify LHS and RHS

LHS
(3.7*10^a)*(5.8*10^b)= 3.7*5.8*10^a+b
=21.46*10^a+b=2146*10^a+b-2

RHS
0.2146*10^c=2146*10^c-4

now equate the power since other terms are equal

a+b-2=c-4
a+b-2+4=c
a+b+2=c

Option B is correct

Bunuel

If $(3.7 * 10^a) * (5.8 * 10^b) = 0.2146 * 10^c$, what is c in terms of a and b?

A. a + b - 2
B. a + b + 2
C. (a + b)/2
D. ab - 2
E. ab + 2

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monsss

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Answer B

(3.7 * 10^a)(5.8 *10^b) = 0.2146*10^c
3.7 * 5.8 * 10^(a+b) = 0.2146 * 10^c
21.46 * 10^(a+b) = 0.2146 * 10^c
0.2146 * 10 ^(a+b+2) = 0.2146 * 10^c

a+ b +2 = c