Solved on Jan 29, 2024

Find the values of the variables $d$ , $e$ , and $f$ that make the given equations true: $\left(2^{d}\right)^{4}=2^{12}$ , $3^{5} \cdot 7^{5}=e^{5}$ , and $5^{0} \cdot 5^{f}=5^{4}$ .

STEP 1

Assumptions
1. We are given three separate equations to solve for different variables: $d$ , $e$ , and $f$ .
2. The base of the exponents in each equation is a prime number.
3. We will use the properties of exponents to solve each equation.

STEP 2

Solve for $d$ in the equation $\left(2^{d}\right)^{4}=2^{12}$ .

STEP 3

Use the property of exponents that states $(a^{m})^{n} = a^{mn}$ to simplify the left side of the equation.
$\left(2^{d}\right)^{4} = 2^{4d}$

STEP 4

Set the exponents on both sides of the equation equal to each other since the bases are the same and the equation must hold true for all values.
$4d = 12$

STEP 5

Divide both sides of the equation by 4 to solve for $d$ .
$d = \frac{12}{4}$

STEP 6

Calculate the value of $d$ .
$d = 3$

STEP 7

Now, solve for $e$ in the equation $3^{5} \cdot 7^{5}=e^{5}$ .

STEP 8

Use the property of exponents that states $a^{m} \cdot b^{m} = (ab)^{m}$ to combine the bases on the left side of the equation.
$3^{5} \cdot 7^{5} = (3 \cdot 7)^{5}$

STEP 9

Simplify the multiplication inside the parentheses.
$(3 \cdot 7)^{5} = 21^{5}$

STEP 10

Set the bases equal to each other since the exponents are the same and the equation must hold true for all values.
$e^{5} = 21^{5}$

STEP 11

Since the exponents are the same, we can equate the bases directly.
$e = 21$

STEP 12

Lastly, solve for $f$ in the equation $5^{0} \cdot 5^{f}=5^{4}$ .

STEP 13

Use the property of exponents that states $a^{0} = 1$ to simplify the left side of the equation.
$5^{0} \cdot 5^{f} = 1 \cdot 5^{f}$

STEP 14

Simplify the multiplication.
$1 \cdot 5^{f} = 5^{f}$

STEP 15

Set the exponents on both sides of the equation equal to each other since the bases are the same and the equation must hold true for all values.
$f = 4$
The values of the variables that make each equation true are $d = 3$ , $e = 21$ , and $f = 4$ .

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Find the values of the variables $d$ , $e$ , and $f$ that make the given equations true: $\left(2^{d}\right)^{4}=2^{12}$ , $3^{5} \cdot 7^{5}=e^{5}$ , and $5^{0} \cdot 5^{f}=5^{4}$ .

STEP 1

Assumptions
1. We are given three separate equations to solve for different variables: $d$ , $e$ , and $f$ .
2. The base of the exponents in each equation is a prime number.
3. We will use the properties of exponents to solve each equation.

STEP 2

Solve for $d$ in the equation $\left(2^{d}\right)^{4}=2^{12}$ .

STEP 3

Use the property of exponents that states $(a^{m})^{n} = a^{mn}$ to simplify the left side of the equation.
$\left(2^{d}\right)^{4} = 2^{4d}$

STEP 4

Set the exponents on both sides of the equation equal to each other since the bases are the same and the equation must hold true for all values.
$4d = 12$

STEP 5

Divide both sides of the equation by 4 to solve for $d$ .
$d = \frac{12}{4}$

STEP 6

Calculate the value of $d$ .
$d = 3$

STEP 7

Now, solve for $e$ in the equation $3^{5} \cdot 7^{5}=e^{5}$ .

STEP 8

Use the property of exponents that states $a^{m} \cdot b^{m} = (ab)^{m}$ to combine the bases on the left side of the equation.
$3^{5} \cdot 7^{5} = (3 \cdot 7)^{5}$

STEP 9

Simplify the multiplication inside the parentheses.
$(3 \cdot 7)^{5} = 21^{5}$

STEP 10

Set the bases equal to each other since the exponents are the same and the equation must hold true for all values.
$e^{5} = 21^{5}$

STEP 11

Since the exponents are the same, we can equate the bases directly.
$e = 21$

STEP 12

Lastly, solve for $f$ in the equation $5^{0} \cdot 5^{f}=5^{4}$ .

STEP 13

Use the property of exponents that states $a^{0} = 1$ to simplify the left side of the equation.
$5^{0} \cdot 5^{f} = 1 \cdot 5^{f}$

STEP 14

Simplify the multiplication.
$1 \cdot 5^{f} = 5^{f}$

STEP 15

Set the exponents on both sides of the equation equal to each other since the bases are the same and the equation must hold true for all values.
$f = 4$
The values of the variables that make each equation true are $d = 3$ , $e = 21$ , and $f = 4$ .

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Find the values of the variables ddd, eee, and fff that make the given equations true: (2d)4=212\left(2^{d}\right)^{4}=2^{12}(2d)4=212, 35⋅75=e53^{5} \cdot 7^{5}=e^{5}35⋅75=e5, and 50⋅5f=545^{0} \cdot 5^{f}=5^{4}50⋅5f=54.

STEP 1

Assumptions1. We are given three separate equations to solve for different variables: ddd, eee, and fff.2. The base of the exponents in each equation is a prime number.3. We will use the properties of exponents to solve each equation.

STEP 2

Solve for ddd in the equation (2d)4=212\left(2^{d}\right)^{4}=2^{12}(2d)4=212.

STEP 3

Use the property of exponents that states (am)n=amn(a^{m})^{n} = a^{mn}(am)n=amn to simplify the left side of the equation.(2d)4=24d\left(2^{d}\right)^{4} = 2^{4d}(2d)4=24d

STEP 4

Set the exponents on both sides of the equation equal to each other since the bases are the same and the equation must hold true for all values.4d=124d = 124d=12

STEP 5

Divide both sides of the equation by 4 to solve for ddd.d=124d = \frac{12}{4}d=412​

STEP 6

Calculate the value of ddd.d=3d = 3d=3

STEP 7

Now, solve for eee in the equation 35⋅75=e53^{5} \cdot 7^{5}=e^{5}35⋅75=e5.

STEP 8

Use the property of exponents that states am⋅bm=(ab)ma^{m} \cdot b^{m} = (ab)^{m}am⋅bm=(ab)m to combine the bases on the left side of the equation.35⋅75=(3⋅7)53^{5} \cdot 7^{5} = (3 \cdot 7)^{5}35⋅75=(3⋅7)5

STEP 9

Simplify the multiplication inside the parentheses.(3⋅7)5=215(3 \cdot 7)^{5} = 21^{5}(3⋅7)5=215

STEP 10

Set the bases equal to each other since the exponents are the same and the equation must hold true for all values.e5=215e^{5} = 21^{5}e5=215

STEP 11

Since the exponents are the same, we can equate the bases directly.e=21e = 21e=21

STEP 12

Lastly, solve for fff in the equation 50⋅5f=545^{0} \cdot 5^{f}=5^{4}50⋅5f=54.

STEP 13

Use the property of exponents that states a0=1a^{0} = 1a0=1 to simplify the left side of the equation.50⋅5f=1⋅5f5^{0} \cdot 5^{f} = 1 \cdot 5^{f}50⋅5f=1⋅5f

STEP 14

Simplify the multiplication.1⋅5f=5f1 \cdot 5^{f} = 5^{f}1⋅5f=5f

STEP 15

Set the exponents on both sides of the equation equal to each other since the bases are the same and the equation must hold true for all values.f=4f = 4f=4The values of the variables that make each equation true are d=3d = 3d=3, e=21e = 21e=21, and f=4f = 4f=4.

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Find the values of the variables $d$ , $e$ , and $f$ that make the given equations true: $\left(2^{d}\right)^{4}=2^{12}$ , $3^{5} \cdot 7^{5}=e^{5}$ , and $5^{0} \cdot 5^{f}=5^{4}$ .

Assumptions
1. We are given three separate equations to solve for different variables: $d$ , $e$ , and $f$ .
2. The base of the exponents in each equation is a prime number.
3. We will use the properties of exponents to solve each equation.

Solve for $d$ in the equation $\left(2^{d}\right)^{4}=2^{12}$ .

Use the property of exponents that states $(a^{m})^{n} = a^{mn}$ to simplify the left side of the equation.
$\left(2^{d}\right)^{4} = 2^{4d}$

Set the exponents on both sides of the equation equal to each other since the bases are the same and the equation must hold true for all values.
$4d = 12$

Divide both sides of the equation by 4 to solve for $d$ .
$d = \frac{12}{4}$

Calculate the value of $d$ .
$d = 3$

Now, solve for $e$ in the equation $3^{5} \cdot 7^{5}=e^{5}$ .

Use the property of exponents that states $a^{m} \cdot b^{m} = (ab)^{m}$ to combine the bases on the left side of the equation.
$3^{5} \cdot 7^{5} = (3 \cdot 7)^{5}$

Simplify the multiplication inside the parentheses.
$(3 \cdot 7)^{5} = 21^{5}$

Set the bases equal to each other since the exponents are the same and the equation must hold true for all values.
$e^{5} = 21^{5}$

Since the exponents are the same, we can equate the bases directly.
$e = 21$

Lastly, solve for $f$ in the equation $5^{0} \cdot 5^{f}=5^{4}$ .

Use the property of exponents that states $a^{0} = 1$ to simplify the left side of the equation.
$5^{0} \cdot 5^{f} = 1 \cdot 5^{f}$

Simplify the multiplication.
$1 \cdot 5^{f} = 5^{f}$

Set the exponents on both sides of the equation equal to each other since the bases are the same and the equation must hold true for all values.
$f = 4$
The values of the variables that make each equation true are $d = 3$ , $e = 21$ , and $f = 4$ .