Exponents Flashcards
Exponents of Base 0 and 1:
0 raised to any power equals 0.
1 raised to any power equals 1.
Value of x in x = x^2:
x must be either 0 or 1.
Decimals and Exponents:
Just like fractions, the value of decimals decreases as exponents increase.
E.g.
- 6^2 = 0.36
- 5^4 = 0.065
- 1^5 = 0.00001
Exponents of Products:
Just as you distribute exponents to both, the denominator and numerator in fractions, you can distribute exponents to products:
E.g.
10^3 = (2x5)^3 = 2^3 x 5^3 = 8x125 = 1000
Works with variables too:
(3x)^4 = 3^4 x x^4 = 81x^4
Negative Base and Exponents:
Unless the negative sign is inside parentheses, the exponent does not distribute.
E.g.
-2^4 = -1 x 2^4 = -1 x 16 = -16
(-2)^4 = (-1)^4 x 2^4 = 1 x 16 = 16
Multiplying Exponential Terms with Same Base:
This only applies when base is the same. This rule is related to fact that exponents are shorthand for repeated multiplication (2^3 = 2x2x2)
When multiplying two exponential terms with same base ADD THE EXPONENTS.
z^2 x z^3 = (zxz) (zxzxz) = z^2+3 = z^5
4 x 4^2 = 4^3
(1/2)^2 x (1/2)^4 = (1/2)^6
Dividing Exponential Terms with Same Base:
When dividing two exponential terms with same base, subtract the exponents.
E.g.
5^6/5^2 = 5^6-2 = 5^4
x^15/x^3 = x^15-3 = x^12
x^3/x^3 = x^3-3 = x^0 = 1 (raised to power of 0 will always give = 1)
Negative Exponent:
Something with a negative exponent is “one over” that same thing with a positive exponent. Take the reciprocal and drop the negative sign.
E.g.
y^-3 = 1/y^3
1/3^-3 = 3^3
(x/4)^-2= (4/x)^2 = 4^2/x^2
Exponents of Exponents:
When you raise an exponential term to an exponent, multiply the exponents.
(a^5)^4 = a^5x4 = a^20
Fractions and Exponents:
The effect of raising a fraction to a power varies depending upon the fraction’s value, the sign, and the exponent. There are four broad categories.
- Even Exponents:
Less than -1:
(-3/2)^2 = 9/4 - Result is bigger
Between -1 and 0:
(-1/2)^2 = 1/4 - Result is bigger
Between 0 and 1:
1/2^2 = 1/4 - Result is smaller
Greater than 1:
3/2^2 = 9/4 - Result is bigger
- Odd Exponents:
Less than -1:
(-3/2)^3 = -27/8 - Result is smaller
Between -1 and 0:
(-1/2)^3 = -1/8 - Result is bigger
Between 0 and 1:
(1/2)^3 = 1/8 - Result is smaller
Greater than 1:
3/2^3 = 27/8 - Result is bigger
Raising Fraction to Negative Power:
To raise a fraction to a negative power, raise the reciprocal to the equivalent positive power.
E.g. (3/7)^-2 = (7/3)^2 = 49/9
Factoring Out Common Term in Exponential Terms:
Normally exponential terms that are added or subtracted cannot be combined. But if two terms with same base are added or subtracted, you can factor out a common term.
E.g. 11^3 + 11^4 = 11^3 (11^0 + 11^1) = 11^3 (1 + 11) = 11^3 x 12
REMEMBER: On GMAT it generally pays to factor exponential terms that have same base in common. That’s generally the way to get to solution in these questions.
E.g.
If x = 4^20 + 4^21 + 4^22, what is the largest prime factor of x?
To find prime factors you need to express x as product. So Factor smallest exponential term out, here 4^20:
4^20 (4^0 + 4^1 + 4^2) = 4^20 (1 + 4 + 16) = 4^20 x 21 =
4^20 x (3x7)
This shows you that the largest prime factor of x is 7.
Equations With Exponents:
- Any number raised to an even exponent becomes positive:
E.g.
3^2 = 9 AND (-3)^2 = 9
x^2 = 25 and |x| = 5 share the same solutions because in both x can be either -5 or 5.
That’s because there is an important relationship: For ANY x, √x^2 = |x|
Another example:
a^2 – 5 = 12
a^2 = 17
The equation has two solutions because a could be positive or negative and both squared would give 17. So, the two solutions for a are:
√17 and -√17
- Odd Exponents keep the sign of base:
Equations that involve only odd exponents have only one solution:
x^3 = -125 Here x can only be -5. Positive value, 5, wouldn’t lead to solution -125.
243 = y^5 Here y has only one solution, 3. The negative value, -3, would yield a negative result and therefore not work.
But if an equations has both odd and even exponents, there are likely to be two solutions.
Examples of Calculating Exponents:
- (3^4)^13 = 3^52
- [(3^30)^12]1/10 = (3^360)^1/10 = 3^360/10 = 3^36
- 3^30 + 3^30 + 3^30 = 3^30 (1+1+1) = 3 x 3^30
- 4 x (3^51) cannot be simplified further.
- (3^100)^1/2 = 3^50
- (4^y + 4^y + 4^y + 4^y) (3^y + 3^y + 3^y )
= 4 x 4^y x 3 x 3^y = 4^y+1 x 3^y+1 = 12^y+1 - 4^a + 4^a+1 = 4^a+2 - 176
In these type of questions you have to express all the exponential terms in terms of the greatest common factor of the terms. In this case it’s 4^a. Use the addition rule to get this:
176 = 4^a+2 - 4^a+1 - 4^a
176 = 4^a x 4^2 - 4^a x 4^1 - 4^a now factor out 4^a
176 = 4^a (4^2 - 4^1 - 1)
176 = 4^a (16 - 4 - 1)
176 = 4^a (11) now divide by 11
16 = 4^a
Therefore a = 2
- If m and n are positive integers what is the value of m in (2)^18 x 5^m = 20^n
Here it’s crucial to recognize that as long as the exponents are integers both sides in this equation must have the same prime factors. So, break the bases down in prime factors and then set exponents equal.
(2)^18 x 5^m = 20^n
(2)^18 x 5^m = (2x2x5)^n
(2)^18 x 5^m = 2^n x 2^n x 5^n
(2)^18 x 5^m = 2^n+n x 5^n
(2)^18 x 5^m = 2^2n x 5^n
This means: 18 = 2n, 9 = n; and m = n
You can conclude: m = 9.
- (1/3)^-4 (1/9)^-3 (1/27)^-2 = ? Again, break down into prime factors and simplify:
= 3^4 x 9^3 x 27^2
= 3^4 x (3x3)^3 x (3x3x3)^2
= 3^4 x (3^2)^3 x (3^3)^2
= 3^4 x 3^6 x 3^6
= 3^4+6+6
= 3^16
Square Root of Quadratic Equations:
If you have something like this:
(z+3)^2 = 25
and you want to find the values for z you can take the square root because you have a perfect square, 25, on the other side. Just remember to consider both scenarios, negative and positive.
√(z+3)^2 = √25
z + 3 = 5 or z + 3 = -5
z = 2 or z = -8
