Chapter 5 Exponents, Bases, +/- Exponents, Fractional Exponent, Simplify, Common Errors,

The base of an exponent expression may be either positive or negative. With a negative base, simply multiple the negative number as many times as the exponent requires.

(-4)^2=16. (-4)^3=-64

Just like proper fractions, decimals btw 0 and 1 decrease as their exponent increases.

(0.6)^2=0.36. (0.5)^4=0.0625. (0.1)^5=0.00001 Remember 1x10^-5

When the base of an exponential expression is a positive proper fraction (in other words, a fraction btw 0 and 1): as the exponent increases, the value of the expression decreases.

(3/4)^1=3/4. (3/4)^2=9/16. (3/4)^3=27/64. 3/4> 9/16> 27/64

You cannot do this with a sum. You must add the numbers inside the parentheses.

Exp. (2+5)^3=7^3=343 (2+5)^3 (not equal) 2^3 + 5^3

1. x^a (X) x^b = x^a+b, Exp. c^3 x c^5 = c^8 5(5^n) = 5^1(5^n) = 5^n+1

2. a^x (X) b^x = ab^x, Exp. 2^4 x 3^4 = 6^4 12^5 = 2^10 x 3^5 12^5 = (2^2x3)^5 distribute the 5 12^5 = 2^10 x 3^5

Factoring Exponential Expressions 7^4 + 7^6, factor our 7^4, 7^4(7^2+1) = 7^4(50)

3^4 + 12^4, factor 12^4 = (2x2x3)^4. Base 3^4, 3^4(1+4^4)

3. x^a/x^b = x(a-b), Exp. 2^5/2^11 = 1/2^6 = 2^-6 x^10/x^3 = x^7

4. (a/b)^x = a^x/b^x, Exp. 10/2^6 = 10^6/2^6 = 5^6 3^5/9^5 = (3/9)^5 = 1/3^5

5. (a^x)^y = a^xy = (a^y)^x, Exp. (3^2)^4 = 3^2x4 = 3^8 3^4x2 = (3^4)^2

6. x^-a = 1/x^a, Exp. (3/2)^-2 = (2/3)^2 = 4/9 2x^-4 = 2/x^4

7. x^a/b = b sq root x^a = (b^sq root x)^a Exp. 27^4/3 = 3^root27^4 = (3^root 27)^4 = 81 (3^root = cube root)

8. a^x + a^x + a^x = 3a^3, Exp. 3^4 + 3^4 + 3^4 = 3x3^4 = 3^5 3^x + 3^x + 3^x=3 X 3^x = 3^x+1

When can you Simplify Exponential Expressions: 2. You can simplify Exponential Expressions linked by multiplication or division if they have either a base or an exponent in common.

CANNOT Simplify. 7^4 + 7^6. 6^"5 - 6^3. 12^7 - 3^7. CAN Simplify (7^4)(7^6) 6^5 / 6^3 12^7 / 3^7

Incorrect. (x+y)^2 = x^2+y^2. (3+2)^2 = 3^2+2^2=13? a^x (X) b^y = (ab)^x+y? 2^4x3^5 = (2x3)^4+5?

Correct (x+y)^2 = x^2+2xy+y^2 (3+2)^2 = 3^2+2(3)(2)+2^2=25 Cannot be simplified further (different bases & different exponents)

Incorrect. axb^x = (axb)^x? 2x3^4 = (2x3)^4?

Correct Cannot be simplified

Incorrect. a x a^x = a^2x? 5 x 5^z = 25^z? -x^2 = x^2? -x^2 -4^2 = 16? -4^2 = -16

Correct a x a^x = a^x+1 5 x 5^z = 5^z+1 cannot be simplified Compare: (-x)^2 = x^2 and (-4)^2 = 16

Incorrect. a^x (X) a^y = a^ay. 5^4 x 5^3 = 5^12? (a^x)^y = ay? (7^4)^3 = 7^7?

Correct a^x (X) a^y = a^x+y 5^4 x 5^3 = 5^7 (a^x)^y = a^xy (7^4)^3 = 7^12

Incorrect. a^x + a^x = a^2x? x^3 + x^2 = x^5. a^x + a^x = a^2x? 2^x + 2^x = 2^2x?

Correct a^x + a^x = 2a^x Cannot be simplified further (addition and different exponents) a^x + a^x = 2a^x 2^x + 2^x = 2(2^x) = 2^x+ 1

When the base of an exponential expression is a product, we can multiply the base together and then raise it to the exponent.

Exp. (2x5)^3=(10)^3=1000 or 2^3x5^3=8x125=1000

When you see a negative exponent, think reciprocal.

Exp. (3/4)^-3 = (4/3)^3 = 64/27

RULE: When raising a power to a power, combine exponents by multiplying.

Exp. (3^2)4 = (3)^2x4 = 3^8

Within the exponent, the numerator tells us what power to raise the base to, and the denominator tells us which root to take. You can raise the base to the power and take root in EITHER order.

Exp. 25^3/2; 3=power, 3=root sq root 25^3 = sq root (5^2)^3 = 5^3 = 125

Any base raised to the exponent of 1 keeps the original base. Any number that does not have an exponent implicitly has an exponent of 1.

Exp. 3^1 = 3, (-6)^1 = -6

RULE: When multiplying two terms with the same base, combine exponents by adding.

Exp. 3^4 x 3^2=3^6

RULE: When dividing two terms with the same base, combine exponents by subtracting.

Exp. 3^6 / 3^2 = 3^(6-2)= 3^4