EXPONENTS AND SQUARE ROOTS PRACTICE FOR SAT

Problem 1 :

If a^-1/2 = 3, what is the value of a ?

a)  -9    b)  1/9     c)  1/3      d)  9

Solution :

a^-1/2 = 3

Taking square on both sides.

(a^-1/2)² = 3²

a^-1 = 9

1/a = 9

a = 1/9

So, the value of a is 1/9.

Problem 2 :

Let n = 1²  + 1⁴ + 1⁶ + 1⁸ + ........ + 1⁵⁰

what is the value of n ?

a)  10     b)  20    c)  25      d)  30

Solution :

n = 1²  + 1⁴ + 1⁶ + 1⁸ + ........ + 1⁵⁰

= 2 + 4 + 6 + 8 + .......... + 50

= 2(1 + 2  + 3 +  ............ + 25)

From this, it is clear the sequence had 25 terms.

So, the sum of the given series is 25.

Problem 3 :

If 4^{2n + 3} = 8^{n + 5}

what is the value of n ?

a)  6     b)  7    c)  8      d)  9

Solution :

4^{2n + 3} = 8^{n + 5}

2^{2(2n + 3)}= 2^{3(n + 5)}

2^{(4n + 6)}= 2^{3n + 15}

Equating the powers, we get

4n + 6 = 3n + 15

4n - 3n = 15 - 6

n = 9

Problem 4 :

If 2^x/2^y = 2³, then x must equal

a)  y + 3     b)  y - 3      c)  3 - y       d)  3y

Solution :

2^x/2^y = 2³

2^{x - y} = 2³

Bases are equal, then we can equate the powers.

x - y = 3

x = 3 + y

Problem 5 :

If 3^x = 10, what is the value of 3^{x - 3} ?

a)  10/3     b)  10/9      c)  10/27       d)  27/10

Solution :

3^{x - 3} = 3^x ⋅3^-3

= 3^x /3³

Applying the value of 3^x, we get

= 10/27

So, option c is correct.

Problem 6 :

If x² y³ = 10 and x³y² = 8, what is the value of x⁵y⁵ ?

a)  18    b)  20      c)  40       d) 80

Solution :

x² y³ = 10 ------(1)

x³y² = 8  ------(2)

(1) ⋅ (2) 

x² y³ ⋅ x³y² = 10(8)

x²⁺³ y³⁺² = 80

x⁵y⁵ = 80

Problem 7 :

If a and b are positive even integers, which of the following is greatest ?

a)  (-2a)^b       b)  (-2a)^2b       c)  (2a)^b        d)  2a^2b

Solution :

Since a and b are even integers, let us take an assumption.

a = 2 and b =4

So, option b is correct.

Problem 8 :

Which of the following is equivalent to x^2a/b, for all values of x ?

Solution :

Problem 9 :

If x² = y³, for what value of z does x^3z = y⁹ ?

a)  -1       b)  0       c)  1        d)  2

Solution :

x² = y³

Raise power 3 on both sides.

(x²)³ = (y³)³

x⁶ = y⁹

Comparing with x^3z = y⁹

3z = 6

z = 6/3

z = 2

So, option d is correct.

Problem 10 :

If 2^x+3 - 2^x = k(2^x), what is the value of k ?

a)  3       b)  5       c)  7        d)  8

Solution :

2^x+3 - 2^x = k(2^x)

2^x ⋅ 2³ - 2^x = k(2^x)

Factoring 2^x, we get

2^x (2³ - 1) = k(2^x)

7 = k

Problem 11 :

If 

a)  1/2       b)  3/4       c)  1        d)  4/3

Solution :

So, a = 1/2.

Problem 12 :

2(√x + 2) = 3√2

If x > 0 in the equation above, what is the value of x ?

a)  2.5    b)  3    c)  3.5      d)  4

Solution :

2(√x + 2) = 3√2

Take square on both sides.

4(x + 2) = 9(2)

4x + 8 = 18

4x = 18 - 8

4x = 10

x = 10/4

x = 5/2

x = 2.5

So, the answer is option a.

Problem 13 :

If x^ac  ⋅ x^bc = x³⁰, x > 1 and a + b = 5, what is the vale of c?

a)  3    b)  5    c)  6      d)  10

Solution :

x^ac  ⋅ x^bc = x³⁰

x^{(ac + bc)} = x³⁰

Equating the powers, we get

ac + bc = 30

Factoring c, we get

c(a + b) = 30

c (5) = 30

c = 30/5

c = 6

Problem 14 :

If n³ = x and n⁴ = 20x, where n > 0, what is the value of x ?

Solution :

n³ = x  ----(1)

n⁴ = 20x ------(2)

(2) / (1)

n⁴ / n³ = 20x / x

n = 20

Applying the value of n in (1), we get

20³ = x

x = 8000

Problem 15 :

If x⁸ y⁷ = 333 and x⁷y⁶ = 3, what is the value of xy ?

Solution :

x⁸ y⁷ = 333 -----(1)

x⁷y⁶ = 3 ------(2)

(1) / (2)

x⁸ y⁷ / x⁷y⁶ = 333/3

xy = 111