Math / Algebra

QuestionHence, or otherwise, solve the following equation, where $x \in \mathbb{R}$ : $2^{4 x+1}=128$

Studdy Solution

STEP 1

1. The equation $2^{4x+1} = 128$ is exponential.
2. We will need to express 128 as a power of 2 to solve for $x$ .

STEP 2

1. Express 128 as a power of 2.
2. Equate the exponents.
3. Solve for $x$ .

STEP 3

First, express 128 as a power of 2. We know that:
$128 = 2^7$
Thus, rewrite the equation as:
$2^{4x+1} = 2^7$

STEP 4

Since the bases are the same, we can equate the exponents:
$4x + 1 = 7$

STEP 5

Solve for $x$ by first subtracting 1 from both sides:
$4x + 1 = 7$ $4x = 7 - 1$ $4x = 6$
Then, divide both sides by 4:
$x = \frac{6}{4}$ $x = \frac{3}{2}$
The value of $x$ is:
$\boxed{\frac{3}{2}}$

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QuestionHence, or otherwise, solve the following equation, where $x \in \mathbb{R}$ : $2^{4 x+1}=128$

Studdy Solution

STEP 1

1. The equation $2^{4x+1} = 128$ is exponential.
2. We will need to express 128 as a power of 2 to solve for $x$ .

STEP 2

1. Express 128 as a power of 2.
2. Equate the exponents.
3. Solve for $x$ .

STEP 3

First, express 128 as a power of 2. We know that:
$128 = 2^7$
Thus, rewrite the equation as:
$2^{4x+1} = 2^7$

STEP 4

Since the bases are the same, we can equate the exponents:
$4x + 1 = 7$

STEP 5

Solve for $x$ by first subtracting 1 from both sides:
$4x + 1 = 7$ $4x = 7 - 1$ $4x = 6$
Then, divide both sides by 4:
$x = \frac{6}{4}$ $x = \frac{3}{2}$
The value of $x$ is:
$\boxed{\frac{3}{2}}$

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QuestionHence, or otherwise, solve the following equation, where x∈Rx \in \mathbb{R}x∈R : 24x+1=1282^{4 x+1}=12824x+1=128

Studdy Solution

STEP 1

1. The equation 24x+1=128 2^{4x+1} = 128 24x+1=128 is exponential.2. We will need to express 128 as a power of 2 to solve for x x x.

STEP 2

1. Express 128 as a power of 2.2. Equate the exponents.3. Solve for x x x.

STEP 3

First, express 128 as a power of 2. We know that:128=27 128 = 2^7 128=27Thus, rewrite the equation as:24x+1=27 2^{4x+1} = 2^7 24x+1=27

STEP 4

Since the bases are the same, we can equate the exponents:4x+1=7 4x + 1 = 7 4x+1=7

STEP 5

Solve for x x x by first subtracting 1 from both sides:4x+1=7 4x + 1 = 7 4x+1=7 4x=7−1 4x = 7 - 1 4x=7−1 4x=6 4x = 6 4x=6Then, divide both sides by 4:x=64 x = \frac{6}{4} x=46​ x=32 x = \frac{3}{2} x=23​ The value of x x x is:32 \boxed{\frac{3}{2}} 23​​

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QuestionHence, or otherwise, solve the following equation, where $x \in \mathbb{R}$ : $2^{4 x+1}=128$

1. The equation $2^{4x+1} = 128$ is exponential.
2. We will need to express 128 as a power of 2 to solve for $x$ .

1. Express 128 as a power of 2.
2. Equate the exponents.
3. Solve for $x$ .

First, express 128 as a power of 2. We know that:
$128 = 2^7$
Thus, rewrite the equation as:
$2^{4x+1} = 2^7$

Since the bases are the same, we can equate the exponents:
$4x + 1 = 7$

Solve for $x$ by first subtracting 1 from both sides:
$4x + 1 = 7$ $4x = 7 - 1$ $4x = 6$
Then, divide both sides by 4:
$x = \frac{6}{4}$ $x = \frac{3}{2}$
The value of $x$ is:
$\boxed{\frac{3}{2}}$