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Big Ideas Math Integrated I, 2016
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Big Ideas Math Integrated I, 2016 View details
4. Solving Exponential Equations
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Exercise 11 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. Note that 64 cannot be represented as a natural power of 16. Therefore, we will write both 64 and 16 as the powers of 4.

64^(2x+4) = 16^(5x)
WritePow

Write as a power

( 4^3 )^(2x+4) = ( 4^2 )^(5x)
PowPow

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

4^(3(2x+4)) = 4^(2*5x)
Distr

Distribute 3

4^(6x+12) = 4^(2*5x)
Multiply

Multiply

4^(6x+12) = 4^(10x)

Now we have two equivalent expressions with the same base. If both sides of the equation are equal, the exponents must also be equal. \

begin{gathered} 4^{6x+12} = 4^{10x} \quad \Leftrightarrow \quad 6x+12 = 10x \end{gathered} Finally, we will solve the equation 6x+12 = 10x.

6x+12 = 10x
SubEqn

LHS-10x=RHS-10x

-4x+12 = 0
SubEqn

LHS-12=RHS-12

-4x = -12
DivEqn

.LHS /(-4).=.RHS /(-4).

x=-12/-4
DivNegNeg

- a/- b=a/b

x=12/4
CalcQuot

Calculate quotient

x=3

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

64^(2x+4) = 16^(5x)
Substitute

x= 3

64^(2( 3)+4) ? = 16^(5( 3))
â–¼
Simplify
Multiply

Multiply

64^(6+4) ? = 16^(15)
AddTerms

Add terms

64^(10) ? = 16^(15)
WritePow

Write as a power

( 4^3 )^(10) ? = ( 4^2 )^(15)
PowPow

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

4^(3*10) ? = 4^(2*15)
Multiply

Multiply

4^(30) = 4^(30) ✓

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

.

Subchapter links expand_more
arrow_right Communicate Your Answer p.299
4
Communicate Your Answer 5 (Page 299)
arrow_right Monitoring Progress p.300-302
Monitoring Progress 1 (Page 300)
Monitoring Progress 2 (Page 300)
Monitoring Progress 3 (Page 300)
Monitoring Progress 4 (Page 301)
Monitoring Progress 5 (Page 301)
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arrow_right Exercises p.303-304
Exercises 1 (Page 303)
Exercises 2 (Page 303)
Exercises 3 (Page 303)
Exercises 4 (Page 303)
Exercises 5 (Page 303)
Exercises 6 (Page 303)
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Exercises 8 (Page 303)
Exercises 9 (Page 303)
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Exercises 11 (Page 303)
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