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Big Ideas Math Integrated I, 2016

BI

Big Ideas Math Integrated I, 2016 View details

4. Solving Exponential Equations

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Exercise 17 Page 303

Rewrite the terms so that they have a common base.

x=3

Practice makes perfect

To solve the given exponential equation, we will start by rewriting the terms so that they have a common base.

36^(-3x+3) = (1/216)^(x+1)

Write as a power

(6^2)^(-3x+3) = (1/6^3)^(x+1)

FracToNegExponent

1/a^m=a^(- m)

(6^2)^(-3x+3) = (6^(-3))^(x+1)

(a^m)^n=a^(m* n)

6^(2(-3x+3)) = 6^(-3(x+1))

Distribute 2

6^(-6x+6) = 6^(-3(x+1))

Distribute -3

6^(-6x+6) = 6^(-3x-3)

Now, we have two equivalent expressions with the same base. If both sides of the equation are equal, the exponents must also be equal. 6^(-6x+6) = 6^(-3x-3) ⇔ -6x+6 = -3x-3 Finally, we will solve the equation -6x+6 = -3x-3.

-6x+6 = -3x-3

LHS+3x=RHS+3x

-3x+6 = -3

LHS-6=RHS-6

-3x = -9

.LHS /(-3).=.RHS /(-3).

x = -9/-3

- a/- b=a/b

x = 9/3

Calculate quotient

x = 3

To check our answer, we will substitute 3 for x in the given equation.

36^(-3x+3) = (1/216)^(x+1)

x= 3

36^(-3( 3)+3) ? = (1/216)^(3+1)

Multiply

36^(-9+3) ? = (1/216)^(3+1)

Add terms

36^(-6) ? = (1/216)^4

Write as a power

(6^2)^(-6) ? = (1/6^3)^4

FracToNegExponent

1/a^m=a^(- m)

(6^2)^(-6) ? = (6^(-3))^4

(a^m)^n=a^(m* n)

6^(2(-6)) ? = 6^(-3*4)

a(- b)=- a * b

6^(-12) ? = 6^(-3*4)

Multiply

6^(-12) = 6^(-12) ✓

Since substituting 3 for x in the given equation produces a true statement, x=3 is the solution to our equation.

Subchapter links

Communicate Your Answer p.299

Monitoring Progress p.300-302

Exercises p.303-304