Chapter 3 - Exponents

Question 1

What is the Base 0 or 1 of exponents raised to?

Answer 1

Base of 0 or 1

0 raised to ANY power equals 0 Ex: 0¹⁰=0

1 raised to ANY power equals 1 Ex: 1⁵ =1

Thus if x=x², x must be either 0 or 1.

Question 2

What is the relationship between exponent and the value of the expression given a positive proper fractional base?

Answer 2

As the exponent increases, the value of the expression decreases

Ex: (3/4)¹ = 3/4

Ex: (3/4)² = 9/16

** Increasing powers cause positive fractions to decrease.

Question 3

What is a compound base?

Answer 3

Just as an exponent can be distributed to a fraction, it can also be distributed to a product.

10³= (2*5)³ = (2)³ * (5)³= 8*125= 1,000

Question 4

A Base of -1

Answer 4

(-1)¹= -1

(-1)²=(-1)(-1)=1

(-1)³= -1*-1*-1= -1

Generally: (-1)^ODD= -1 (-1)^EVEN= 1

Even Exponents hide the sign of the original number because they will always result in positive value.

​ODD exponents preserve the sign of the original expression

Question 5

A Negative Base?

Answer 5

When dealing with negative bases, pay particular attention to PEMDAS. Unless the negative sign is inside parentheses, the exponent DOES NOT distribute.

     - 2<sup>4</sup>                 *** ****≠**              *   (-2)<sup>4</sup> - 2<sup>4</sup>= -1\*2<sup>4</sup>= - 16                               (-2)<sup>4</sup>=(-1)<sup>4</sup>\* (2)<sup>4</sup>=1 \*16 = 16

Question 6

Combining exponential term with common bases

Answer 6

Multiply terms with same base: Add exponents

EX: z²* z⁴ = z⁶

4¹*4²= 4³

Question 7

Combining exponential term with common bases

Answer 7

Divide Terms with same bases: Subtract Exponents

EX:5⁶/5²= 5⁴

EX: x¹⁵/x⁸= x⁷

Question 8

Anything raised to the zero equals one

Answer 8

If you divide something by itself, the quotient is 1

EX: a³/ a³= a ^3-3 = a⁰= 1

**A Base raised to the 0 power equals 1. **The one exception is a base of 0. 0⁰is undefined. That’s because 0/0 is undefined.

Question 9

Negative Exponents

Answer 9

y²/ y⁵= y ^2-5= y ^-3= 1/y³

G/R: Something with a negative exponent is just “one over” that same thing with a positive exponent

EX: 1/3^-3= 3³

(x/4)^-2 = 4²/ x²

Question 10

Nested Exponents: Multiply Exponents

Answer 10

(z²)³= z⁶

(a⁵)⁴= a^5*4 = a²⁰

Question 11

Fractional Exponents

Answer 11

Fractional exponents are the link between exponents and roots. Num tells you what power to raise the base to, and the Den tells you which root to take.

Ex: 25 ^3/2=√(25)³=√(5²)³ = 5³= 125.

OR

25 ^3/2= (5²) ^3/2 = 5^2*3/2= 5³= 125

Question 12

Factoring out a common term

Answer 12

If two terms with the same base are added or subtracted, you can factor out a common term

**Ex: **11³+ 11⁴= 11³(1+11) = 11³(12)

Question 13

Equations with exponents

Note: Not all even exponents have 2 solutions.

EX: x²= 0 which only has 1 solution = 0.

x² = -9 Squaring can never produce a negative number. This equation does not have any solutions.

Answer 13

Even Exponents hide the sign of the **base **(2 solutions)

3²= 9 AND (-3)²=9

x²= 25 | x | = 5

Imp relationship: For any x, **√(x)² = | x |**

**ODD **Exponents Keep the sign of the **base **( only 1 solution)

x³= - 125 = x = - 5

243 = y⁵= y =3

Question 14

Simplify:

Answer 14

(4^y+4^y+4^y+ 4^y) (3^y+3^y+3^y)

4^y( 1+1+1+1) 3^y(1+1+1)

4^y(4¹) 3^y(3¹)

(4^{y+1) (}3^{y+1) = Same exponent, diff base}

(12)^y+1