Exercise 43 Page 388

Practice makes perfect

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

16^(3/4)=4^x

WritePow

Write as a power

(4^2)^(3/4) = 4^x

PowPow

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

4^(2(3/4)) = 4^x

MoveLeftFacToNum

a*b/c= a* b/c

4^(6/4)=4^x

ReduceFrac

a/b=.a /2./.b /2.

4^(3/2)=4^x

Now we have two equivalent expressions with the same base. If both sides of the equation are equal, the exponents must also be equal. 4^(3/2)=4^x ⇔ 3/2 = x Therefore, x= 32 is the solution to our equation.

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

8^(1/3)=4^x

WritePow

Write as a power

(2^3)^(1/3)=(2^2)^x

PowPow

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

2^(3(1/3))=2^(2x)

DenomMultFracToNumber

3 * a/3= a

2^1=2^(2x)

Now we have two equivalent expressions with the same base. If both sides of the equation are equal, the exponents must also be equal. 2^1=2^2x ⇔ 1=2x Finally, we will solve the equation 1=2x.

1=2x

DivEqn

.LHS /2.=.RHS /2.

1/2=x

RearrangeEqn

Rearrange equation

x=1/2

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

3^(4x) = 9^2

WritePow

Write as a power

3^(4x) = ( 3^2 )^2

PowPow

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

3^(4x) = 3^(2(2))

Multiply

3^(4x) = 3^4

Now we have two equivalent expressions with the same base. If both sides of the equation are equal, the exponents must also be equal. 3^(4x) = 3^4 ⇔ 4x = 4 Finally, we will solve the equation 4x = 4.

4x = 4

DivEqn

.LHS /4.=.RHS /4.

x=1

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

(1/2)^x=4

WritePow

Write as a power

(2^0/2^1)^x=2^2

DivPow

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

(2^(0-1))^x=2^2

SubTerm

Subtract term

(2^(-1))^x=2^2

PowPow

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

2^(- x) = 2^2

Now we have two equivalent expressions with the same base. If both sides of the equation are equal, the exponents must also be equal. 2^(- x) = 2^2 ⇔ - x = 2 Finally, we will solve the equation - x = 2.

- x = 2

MultEqn

LHS * (-1)=RHS* (-1)

x=-2

Subchapter links

7.1.1 Explore–Conjecture–Prove p.376-379

7.1.2 Properties of Rhombi p.382-383

7.1.3 Two-Column Proofs p.387-388

7.1.4 More Geometric Proofs p.390-391

7.1.5 Using Similar Triangles to Prove Theorems p.395-396

Exercise 43 Page 388

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