Question Solve for complex numbers and rational exponents without a calculator. Find the values of the variables a, b, c, d, and f that satisfy the given equations.
1. $x^{2}=16$ solutions: $\{4, -4\}$
2. a. $3^{4} \cdot 3^{2}=3^{a}$ , b. $\frac{5^{4}}{5^{3}}=5^{b}$ , c. $4^{c}=1$ , d. $2^{6} \cdot d^{6}=14^{6}$ , e. $6^{f}=\frac{1}{6}$

Studdy Solution

STEP 1

Assumptions
1. We are working with real numbers and their properties.
2. The laws of exponents apply for rational exponents.
3. No calculator is to be used for the calculations.

STEP 2

For the first question, we need to find all the solutions to $x^{2}=16$ . The solutions are the numbers that, when squared, equal 16.

STEP 3

Check each option by squaring them to see if they equal 16.
For option A: $256^2$ For option B: $8^2$ For option C: $4^2$ For option D: $(-256)^2$ For option E: $(-8)^2$ For option F: $(-4)^2$

STEP 4

Calculate the squares of the options:
A. $256^2$ is much larger than 16, so A is not a solution. B. $8^2 = 64$ , which is not equal to 16, so B is not a solution. C. $4^2 = 16$ , which is equal to 16, so C is a solution. D. $(-256)^2$ is much larger than 16, so D is not a solution. E. $(-8)^2 = 64$ , which is not equal to 16, so E is not a solution. F. $(-4)^2 = 16$ , which is equal to 16, so F is a solution.

STEP 5

The solutions to $x^{2}=16$ are options C and F.

STEP 6

For the second question, we need to find the value of each variable that makes the equation true.
a. $3^{4} \cdot 3^{2}=3^{a}$

STEP 7

Use the law of exponents that states $a^{m} \cdot a^{n} = a^{m+n}$ to combine the exponents on the left side.

STEP 8

Combine the exponents:
$3^{4} \cdot 3^{2} = 3^{4+2} = 3^{6}$

STEP 9

Since the bases are the same, the exponents must be equal for the equation to be true.
$3^{a} = 3^{6}$

STEP 10

Therefore, $a = 6$ .
b. $\frac{5^{4}}{5^{3}}=5^{b}$

STEP 11

Use the law of exponents that states $a^{m} / a^{n} = a^{m-n}$ to simplify the left side.

STEP 12

Simplify the expression:
$\frac{5^{4}}{5^{3}} = 5^{4-3} = 5^{1}$

STEP 13

Since the bases are the same, the exponents must be equal for the equation to be true.
$5^{b} = 5^{1}$

STEP 14

Therefore, $b = 1$ .
c. $4^{c}=1$

STEP 15

Any non-zero number raised to the power of 0 is 1.

STEP 16

Therefore, $c = 0$ because $4^{0} = 1$ .
d. $2^{6} \cdot d^{6}=14^{6}$

STEP 17

Use the law of exponents that states $(a \cdot b)^{n} = a^{n} \cdot b^{n}$ to rewrite the right side.

STEP 18

Rewrite the right side:
$14^{6} = (2 \cdot 7)^{6} = 2^{6} \cdot 7^{6}$

STEP 19

Since the bases of $2^{6}$ are the same on both sides, we can equate the other parts of the equation.
$d^{6} = 7^{6}$

STEP 20

Therefore, $d = 7$ because $d^{6} = 7^{6}$ .
e. $6^{f}=\frac{1}{6}$

STEP 21

Rewrite $\frac{1}{6}$ as $6^{-1}$ .

STEP 22

Since the bases are the same, the exponents must be equal for the equation to be true.
$6^{f} = 6^{-1}$

STEP 23

Therefore, $f = -1$ .
The solutions are:
1. C. 4 and F. -4
2. a. $a = 6$ b. $b = 1$ c. $c = 0$ d. $d = 7$ e. $f = -1$

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Studdy Solution

STEP 1

Assumptions1. We are working with real numbers and their properties.2. The laws of exponents apply for rational exponents.3. No calculator is to be used for the calculations.

STEP 2

For the first question, we need to find all the solutions to x2=16x^{2}=16x2=16. The solutions are the numbers that, when squared, equal 16.

STEP 3

Check each option by squaring them to see if they equal 16.For option A: 2562256^22562 For option B: 828^282 For option C: 424^242 For option D: (−256)2(-256)^2(−256)2 For option E: (−8)2(-8)^2(−8)2 For option F: (−4)2(-4)^2(−4)2

STEP 4

STEP 5

The solutions to x2=16x^{2}=16x2=16 are options C and F.

STEP 6

For the second question, we need to find the value of each variable that makes the equation true.a. 34⋅32=3a3^{4} \cdot 3^{2}=3^{a}34⋅32=3a

STEP 7

Use the law of exponents that states am⋅an=am+na^{m} \cdot a^{n} = a^{m+n}am⋅an=am+n to combine the exponents on the left side.

STEP 8

Combine the exponents:34⋅32=34+2=363^{4} \cdot 3^{2} = 3^{4+2} = 3^{6}34⋅32=34+2=36

STEP 9

Since the bases are the same, the exponents must be equal for the equation to be true.3a=363^{a} = 3^{6}3a=36

STEP 10

Therefore, a=6a = 6a=6.b. 5453=5b\frac{5^{4}}{5^{3}}=5^{b}5354​=5b

STEP 11

Use the law of exponents that states am/an=am−na^{m} / a^{n} = a^{m-n}am/an=am−n to simplify the left side.

STEP 12

Simplify the expression:5453=54−3=51\frac{5^{4}}{5^{3}} = 5^{4-3} = 5^{1}5354​=54−3=51

STEP 13

Since the bases are the same, the exponents must be equal for the equation to be true.5b=515^{b} = 5^{1}5b=51

STEP 14

Therefore, b=1b = 1b=1.c. 4c=14^{c}=14c=1

STEP 15

Any non-zero number raised to the power of 0 is 1.

STEP 16

Therefore, c=0c = 0c=0 because 40=14^{0} = 140=1.d. 26⋅d6=1462^{6} \cdot d^{6}=14^{6}26⋅d6=146

STEP 17

Use the law of exponents that states (a⋅b)n=an⋅bn(a \cdot b)^{n} = a^{n} \cdot b^{n}(a⋅b)n=an⋅bn to rewrite the right side.

STEP 18

Rewrite the right side:146=(2⋅7)6=26⋅7614^{6} = (2 \cdot 7)^{6} = 2^{6} \cdot 7^{6}146=(2⋅7)6=26⋅76

STEP 19

Since the bases of 262^{6}26 are the same on both sides, we can equate the other parts of the equation.d6=76d^{6} = 7^{6}d6=76

STEP 20

Therefore, d=7d = 7d=7 because d6=76d^{6} = 7^{6}d6=76.e. 6f=166^{f}=\frac{1}{6}6f=61​

STEP 21

Rewrite 16\frac{1}{6}61​ as 6−16^{-1}6−1.

STEP 22

Since the bases are the same, the exponents must be equal for the equation to be true.6f=6−16^{f} = 6^{-1}6f=6−1

STEP 23

Therefore, f=−1f = -1f=−1.The solutions are:1. C. 4 and F. -42. a. a=6a = 6a=6 b. b=1b = 1b=1 c. c=0c = 0c=0 d. d=7d = 7d=7 e. f=−1f = -1f=−1

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Assumptions
1. We are working with real numbers and their properties.
2. The laws of exponents apply for rational exponents.
3. No calculator is to be used for the calculations.

For the first question, we need to find all the solutions to $x^{2}=16$ . The solutions are the numbers that, when squared, equal 16.

Check each option by squaring them to see if they equal 16.
For option A: $256^2$ For option B: $8^2$ For option C: $4^2$ For option D: $(-256)^2$ For option E: $(-8)^2$ For option F: $(-4)^2$

The solutions to $x^{2}=16$ are options C and F.

For the second question, we need to find the value of each variable that makes the equation true.
a. $3^{4} \cdot 3^{2}=3^{a}$

Use the law of exponents that states $a^{m} \cdot a^{n} = a^{m+n}$ to combine the exponents on the left side.

Combine the exponents:
$3^{4} \cdot 3^{2} = 3^{4+2} = 3^{6}$

Since the bases are the same, the exponents must be equal for the equation to be true.
$3^{a} = 3^{6}$

Therefore, $a = 6$ .
b. $\frac{5^{4}}{5^{3}}=5^{b}$

Use the law of exponents that states $a^{m} / a^{n} = a^{m-n}$ to simplify the left side.

Simplify the expression:
$\frac{5^{4}}{5^{3}} = 5^{4-3} = 5^{1}$

Since the bases are the same, the exponents must be equal for the equation to be true.
$5^{b} = 5^{1}$

Therefore, $b = 1$ .
c. $4^{c}=1$

Therefore, $c = 0$ because $4^{0} = 1$ .
d. $2^{6} \cdot d^{6}=14^{6}$

Use the law of exponents that states $(a \cdot b)^{n} = a^{n} \cdot b^{n}$ to rewrite the right side.

Rewrite the right side:
$14^{6} = (2 \cdot 7)^{6} = 2^{6} \cdot 7^{6}$

Since the bases of $2^{6}$ are the same on both sides, we can equate the other parts of the equation.
$d^{6} = 7^{6}$

Therefore, $d = 7$ because $d^{6} = 7^{6}$ .
e. $6^{f}=\frac{1}{6}$

Rewrite $\frac{1}{6}$ as $6^{-1}$ .

Since the bases are the same, the exponents must be equal for the equation to be true.
$6^{f} = 6^{-1}$

Therefore, $f = -1$ .
The solutions are:
1. C. 4 and F. -4
2. a. $a = 6$ b. $b = 1$ c. $c = 0$ d. $d = 7$ e. $f = -1$