Exponents and roots

Question 1

What are exponents?

Answer 1

The exponent, or power, tells you how many bases to multiply together.
5³ = 5 x 5 x 5 = 125 [5 is the base and 3 is the exponent]

Question 2

What is an exponential expression or term?

Answer 2

Is the expression or term that contains an exponent and can contain variables as well. The variable can be the base, the exponent, or even both.
a³ = a x a x a

Question 3

What is the result of any base to the first power?

Answer 3

Is always equal to that base.
7¹ = 7
10¹ = 10
1¹ = 1

Question 4

Remember the following powers:
11² = ?
12² = ?
15² = ?
20² = ?
30² = ?
1³ = ?
2³ = ?
3³ = ?
4³ = ?
5³ = ?
10³ = ?

Answer 4

121
144
225
400
900
1
8
27
64
125
1000

Question 5

Is -(3)² = (-3)²?

Answer 5

No. Due to the application of PEMDAS rule, the placement of the negative sign makes a significant difference.
In the first case you square before applying the negative sign, so the answer is negative:
- (3)² = - (3 x 3) = - (9) = - 9
In the second case, you square a negative number, so the answer is positive:
(-3)² = (-3 x -3) = 9

Question 6

When a negative number is raised to an even power, will the result be negative?

Answer 6

No. Negative numbers raised to an even power are always positive.

Question 7

When a negative number is raised to an odd power, will the result be negative?

Answer 7

Yes, negative numbers raised to an odd number are always negative

Question 8

A positive base raised to any power will always be positive. True or false?

Answer 8

True, because positive times positive is positive no matter how many times you multiply.

Question 9

Why is necessary to be careful when looking to an exponential expression or term with an even exponent?

Answer 9

Because an even exponent can hide the sign of the base.
x² = 16
As an even exponent always give a positive result, the answer can be either 4 or -4.
Always be careful when dealing with even exponents in equations. Look for more than one possible solution.

Question 10

What is the result of any base to the power of 0?

Answer 10

The result is almost always 1.

Question 11

What should you do when multiplying terms with same base?

Answer 11

Add the exponents.
a² x a = a³ [treat any term without an exponent as if it had an exponent of 1]

Question 12

What should you do when dividing terms with same base?

Answer 12

Subtract the exponents.
a³ : a = a²

Question 13

What equals to a negative power?

Answer 13

A negative power is always equal to one over a positive power
a^-2 = 1/a²
Therefore any base to a negative power is equal to the reciprocal of the base to the positive power
a^-2 is equal to the reciprocal of a²

Question 14

What is the reciprocal?

Answer 14

The reciprocal of 5 is 1 over 5, or 1/5. Something times its reciprocal always equals to 1:
5 x 1/5 = 1
a² x 1/a² = 1

Question 15

What should you do when you raise something that already has an exponent to another power?

Answer 15

Multiply the two exponents together.
(a²)² = a^4

Question 16

What can you do if you have different bases that are numbers?

Answer 16

Try breaking down the bases to prime factors.
2^2 * 4^3 * 16 =
2^2 * (2^2)^3 * 2^4 =
2^2 * 2^6 * 2^4 =
2^2+6+4 =
2^12

Question 17

What can you do if you apply an exponent to a product?

Answer 17

When you apply an exponent to a product, apply the exponent to each factor in the product
(ab)³ =
ab x ab x ab =
(a x a x a)(b x b x b) =
a³b³
You can also do the reverse by rewriting a³b³ as (ab)³

Question 18

What can you do if you add or subtract terms with the same base?

Answer 18

In that case you’ll need to pull out the common factor:
13²+13³ =
13²(1+13²)
If you were given x’s instead of 13’s as bases, the factoring would work the same way:
x^5 + x³ = x³x² + x³ = x³(x² + 1)

Question 19

What is the root?

Answer 19

Is the process to undo the application of an exponent.
3² = 9 and √9 = 3

Question 20

What do you get if you square-root first and then square?

Answer 20

You get back the original number:
(√16)² = √16 * √16 = 16

Question 21

What do you get if you square first, then square-root?

Answer 21

You get back to the original number if the original number is positive:
√5² = √5 x 5 = √25 = 5
Or you get back to the original number, but with a positive sign, if the base was negative
√(-5)² = √-5 x -5 = √25 = 5

Question 22

What are √2 and √3?

Answer 22

They are not perfect squares, but you can remember the approximation:
√2 =~ 1.4 (February 14, valentine’s day)
√3 =~ 1.7 (March 17, St. Patrick’s day)

Question 23

Is the square root of a number between 0 and 1 greater than the original number?

Answer 23

Yes. When you take the square root of any number greater than 1, your answer will be less than the original number:
√2 < 2
√21 < 21
√1.3 < 1.3
However, the square root of a number between 0 and 1 is greater than the original number:
√0.5 > 0.5 =~ 0.7
√2/3 > 2/3 =~ 0.8

Question 24

For expressions with positive bases, a square root is equivalent to an exponent of 1/2. True or false?

Answer 24

True. You can rewrite a square root as an exponent of 1/2
√7^22 =
(7^22)^1/2 =
7^22/2 =
7^11

Question 25

What is a cube root?

Answer 25

Cube-rooting undoes the process of raising a number to the third power.
4³ = 64 and ³√64 = 4

Question 26

What is the main property difference between square and cube roots?

Answer 26

Is that you can take the cube root of a negative number. You always get a negative number:
³√-64 = - 4 because (- 4)³ = - 64

Question 27

For expressions with positive bases, a cube root is equivalent to an exponent of 1/3. True or false?

Answer 27

True. Therefore, if you have a number raised to a fractional exponent with divisor 3, you can rewrite the expression to find the cube root.
8^2/3 = 8^2 * 1/3 = ³√8^2 = ³√64 = 4
or
8^2/3 = 8^1/3 * 2 = (³√8)² = 2² = 4

Question 28

What can you do if you multiply square roots?

Answer 28

You can just put everything under the same radical sign
√20 x √5 = √20x5 = √100 = 10

Question 29

How do you factor a square root?

Answer 29

If the answer you get isn’t a perfect square itself but does have a perfect square as a factor, then the GMAT will typically expect you to simplify that answer as far as if can go.
√6 x √2 =
√6x2 =
√12 =
√4 x 3 =
√4 x √3 =
2√3

Question 30

What to do if you have an addition or subtraction underneath the square root symbol?

Answer 30

Factor out a square factor from the sum or difference
√10³ - 10² =
√10² (10¹ - 1)=
√10² x 9 =
10 x 3 =
30

Question 31

How to simplify 8^2/3?

Answer 31

(³√8)²
The divisor becomes the root and the dividend becomes the exponent