- C When a question adds or subtracts exponential terms, a good rule of thumb
is to factor out what the terms have in common. 3^7 and 3^5 share 3^5 as a factor,
so the left side of the equation can be rewritten as 3^5 (3^2 − 1) = 3^5 (8) = (3^5 )(2^3 ).
Thus
(3^5 )(2^3 ) = (2x)(3y)
x = 3 y = 5
↓
x + y = 8 - C When multiplying square roots, combine the terms underneath one radical.
Thus √a^3 × √a × b^2 = √a^3 × a × b^2 = √a^4 × b^2 = √a^4 × √b^2 = a^2 × b.
Quantitative Comparison Questions
- A To make the columns comparable, rewrite 25^8 with a base of 5: 25^8 = (5^2 )^8 =
516. Now that both columns are expressed in base 5, the column with the larger
exponent will have the greater value. The exponent in Quantity A is greater, so
the answer is A. - C Simplify Quantity A: aa–4–3 = a−3 −(− 4) = a−3+4 = a^1 = a. The two quantities are
equal. - C Simplify Quantity A by expressing the numerator with base 2: (2^2 )−
v
2 −w =
2 −2v
2 −w =
2 −2v−(−w) = 2−2v+w. Substitute 2v for w: 2−2v+2v = 2^0 = 1. The two quantities
are equal.- D Although b > a, b is raised to a smaller exponent than a is. Thus a
relationship cannot be determined. For illustration, plug in numbers: if b =
2 and a = 1, then b^2 = 4 and a^3 = 1. In this case, Quantity A is greater. But if
b = 4 and a = 3, b^2 = 16, and a^3 = 27. In this case, Quantity B is greater. The
relationship cannot be determined. - D First, rewrite all the terms in the given equation to have base 2: (2a)(2^2 b) = 2^6
=> a + 2b = 6. Now you can plug in numbers: if a = 0, then b = 3. In this case, a- b = 3, and the two quantities are equal. If b = 0, then a = 6. In this case, a + b
= 6, and Quantity A is greater. A relationship cannot be determined.
- b = 3, and the two quantities are equal. If b = 0, then a = 6. In this case, a + b
- B When a fraction is raised to a power, the result gets closer to zero as the
exponent increases. Since x is a negative fraction, the result will become less
negative as the exponent increases. Thus when −1 < x < 0, x^5 is less negative
than x^3 , meaning that x^5 > x^3. Quantity B is greater. - A Your initial goal should be to manipulate the columns to make the bases
comparable. However, it is not possible to express base 20 using base 2. Thus a
different approach is required. Instead of making the bases comparable, make
the exponents comparable. Note that 120 = (3)(40). Therefore, 2^120 = 2(3×40) =
(2^3 )^40 = 8^40. Now the two quantities have the same exponent. The quantity with
the greater base will have the greater value.
282 PART 4 ■ MATH REVIEW03-GRE-Test-2018_173-312.indd 282 12/05/17 11:54 am