Exponents and roots Flashcards
What are exponents?
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]
What is an exponential expression or term?
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
What is the result of any base to the first power?
Is always equal to that base.
7¹ = 7
10¹ = 10
1¹ = 1
Remember the following powers:
11² = ?
12² = ?
15² = ?
20² = ?
30² = ?
1³ = ?
2³ = ?
3³ = ?
4³ = ?
5³ = ?
10³ = ?
121
144
225
400
900
1
8
27
64
125
1000
Is -(3)² = (-3)²?
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
When a negative number is raised to an even power, will the result be negative?
No. Negative numbers raised to an even power are always positive.
When a negative number is raised to an odd power, will the result be negative?
Yes, negative numbers raised to an odd number are always negative
A positive base raised to any power will always be positive. True or false?
True, because positive times positive is positive no matter how many times you multiply.
Why is necessary to be careful when looking to an exponential expression or term with an even exponent?
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.
What is the result of any base to the power of 0?
The result is almost always 1.
What should you do when multiplying terms with same base?
Add the exponents.
a² x a = a³ [treat any term without an exponent as if it had an exponent of 1]
What should you do when dividing terms with same base?
Subtract the exponents.
a³ : a = a²
What equals to a negative power?
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
What is the reciprocal?
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
What should you do when you raise something that already has an exponent to another power?
Multiply the two exponents together.
(a²)² = a^4
What can you do if you have different bases that are numbers?
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
What can you do if you apply an exponent to a product?
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
What can you do if you add or subtract terms with the same base?
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)
What is the root?
Is the process to undo the application of an exponent.
3² = 9 and √9 = 3
What do you get if you square-root first and then square?
You get back the original number:
(√16)² = √16 * √16 = 16
What do you get if you square first, then square-root?
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
What are √2 and √3?
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)
Is the square root of a number between 0 and 1 greater than the original number?
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
For expressions with positive bases, a square root is equivalent to an exponent of 1/2. True or false?
True. You can rewrite a square root as an exponent of 1/2
√7^22 =
(7^22)^1/2 =
7^22/2 =
7^11
What is a cube root?
Cube-rooting undoes the process of raising a number to the third power.
4³ = 64 and ³√64 = 4
What is the main property difference between square and cube roots?
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
For expressions with positive bases, a cube root is equivalent to an exponent of 1/3. True or false?
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
What can you do if you multiply square roots?
You can just put everything under the same radical sign
√20 x √5 = √20x5 = √100 = 10
How do you factor a square root?
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
What to do if you have an addition or subtraction underneath the square root symbol?
Factor out a square factor from the sum or difference
√10³ - 10² =
√10² (10¹ - 1)=
√10² x 9 =
10 x 3 =
30
How to simplify 8^2/3?
(³√8)²
The divisor becomes the root and the dividend becomes the exponent
