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Solved Examples: Exponents and Radicals | Mathematics for SAT PDF Download

Q1: If 3x - 3 = 27, what is the value of x?
Solution:

In this question, we are trying to solve for the variable in exponent. In an equation, the idea to solve for the variable in exponent is to get the same base on both sides of the equation.

We have 27 on the right side of the given equation.

27 is a power of 3, that is 27 = 33.

3x - 3 = 27

3x - 3 = 33

In the equation, we have the same base on both sides. So, we can equate the exponents.

x - 3 = 3

Add 3 to both sides.

x = 6

Q2: If 9= 2782, what is the value of x?
Solution:

9= 2782

(32)= (33)82 

32x = 3246

2x = 246

Divide both sides by 2.

x = 123

Q3: If y= m and y9 = 2/3, what is the value of y in terms of m?
Solution:

y= m ----(1)

y9 = 2/3 

----(2)

Divide (2) by (1).

(2) ÷ (1) ----> y÷ yn = m ÷ (2/3)

y9 - 8 = m ⋅ (3/2)

y = 3m/2

Q4: k2x2a = x2a + 2
In the equation above, k, x and a are positive integers greater than 1. What is the value of x - k?
Solution:

k2x2a = x2a + 2

k2x2a = x2ax2

Divide both sides by x2a.

k2 x2

Since both k and x are positive,

k = x

Therefore,

x - k = 0

Q5: Solved Examples: Exponents and Radicals | Mathematics for SAT
In the equation above, x > 1. Find the value of (a - b) such that (a - b) > 0. 
Solution:

Solved Examples: Exponents and Radicals | Mathematics for SAT

(a + b)2 = 25

Taking square root on both sides.

(a + b)2 = 25

a + b = ±5

a + b = 5  or  a + b = -5

Since (a + b) > 0,

a + b = 5

Q6: Which of the following is equal to 3√x⋅ √x?
(a) x
(b) x9/4
(c) x13/6
(d) x3
Ans:
(c)

3√x⋅ √x3 = (x2)1/3 ⋅ (x3)1/2

= x2/⋅ x3/2

= x2/3 + 3/2

= x4/6 + 9/6

= x(4 + 9)/6

= x13/6

The correct answer choice is (c).

Q7: If 3√x√72, what is the value of x?
Solution:

3√x√72

√9√x√72

√(9x3) √72

Squaring both sides,

9x3 72

Divide both sides by 9.

x3 = 8

x3 = 23

x = 2

Q8: If 3x - y = 12, what is the value of 8÷ 2y?
Solution:

8÷ 2y = (23)÷ 2y

= 23÷ 2y

= 23x - y

Substitute 3x - y = 12.

= 212

Q9: Solved Examples: Exponents and Radicals | Mathematics for SAT
In the equation above, if x > 1 and a + b = 2, what is the value of (a - b)?
Solution:

Solved Examples: Exponents and Radicals | Mathematics for SAT

a2 - b2 = 16

(a + b)(a - b) = 16

Substitute a + b = 2.

2(a - b) = 16

Divide both sides by 2.

a - b = 8

Q10: Solved Examples: Exponents and Radicals | Mathematics for SAT
In the expression above x > 1 and y > 1. Which of the following is equivalent to the above expression ?
(a) Solved Examples: Exponents and Radicals | Mathematics for SAT
(b) Solved Examples: Exponents and Radicals | Mathematics for SAT
(c) Solved Examples: Exponents and Radicals | Mathematics for SAT
(d) Solved Examples: Exponents and Radicals | Mathematics for SAT

Ans: (d)

Solved Examples: Exponents and Radicals | Mathematics for SAT

Solved Examples: Exponents and Radicals | Mathematics for SAT

The correct answer choice is (d).

The document Solved Examples: Exponents and Radicals | Mathematics for SAT is a part of the SAT Course Mathematics for SAT.
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