Solve for k 8^(3k+4)=4^(2k-1)

In the given problem, you are to determine the value of the variable 'k' that makes the equation true. The equation provided is an exponential equation where the base '8' is raised to the power of '(3k+4)' on one side, and the base '4' is raised to the power of '(2k-1)' on the other side. You will need to apply properties of exponents and perhaps logarithms to solve for the unknown 'k'.

$8^{3 k + 4} = 4^{2 k - 1}$

Answer

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Solution:

Step 1:

Convert the given equation to have the same base by expressing each side as powers of 2. $2^{3 (3 k + 4)} = 2^{2 (2 k - 1)}$

Step 2:

Equating the exponents because the bases are now identical. $3 (3 k + 4) = 2 (2 k - 1)$

Step 3:

Isolate the variable $k$ .

Step 3.1:

Expand $3 (3 k + 4)$ .

Step 3.1.1:

Introduce zero without changing the equation. $0 + 0 + 3 (3 k + 4) = 2 (2 k - 1)$

Step 3.1.2:

Confirm that adding zero does not change the expression. $3 (3 k + 4) = 2 (2 k - 1)$

Step 3.1.3:

Distribute $3$ over $(3 k + 4)$ . $3 (3 k) + 3 \cdot 4 = 2 (2 k - 1)$

Step 3.1.4:

Perform the multiplication.

Step 3.1.4.1:

Multiply $3$ by $3 k$ . $9 k + 3 \cdot 4 = 2 (2 k - 1)$

Step 3.1.4.2:

Multiply $3$ by $4$ . $9 k + 12 = 2 (2 k - 1)$

Step 3.2:

Expand $2 (2 k - 1)$ .

Step 3.2.1:

Distribute $2$ over $(2 k - 1)$ . $9 k + 12 = 2 (2 k) + 2 \cdot (- 1)$

Step 3.2.2:

Perform the multiplication.

Step 3.2.2.1:

Multiply $2$ by $2 k$ . $9 k + 12 = 4 k + 2 \cdot (- 1)$

Step 3.2.2.2:

Multiply $2$ by $- 1$ . $9 k + 12 = 4 k - 2$

Step 3.3:

Get all terms with $k$ on one side.

Step 3.3.1:

Subtract $4 k$ from both sides. $9 k + 12 - 4 k = - 2$

Step 3.3.2:

Combine like terms. $5 k + 12 = - 2$

Step 3.4:

Move constant terms to the other side.

Step 3.4.1:

Subtract $12$ from both sides. $5 k = - 2 - 12$

Step 3.4.2:

Combine the constants. $5 k = - 14$

Step 3.5:

Divide to solve for $k$ .

Step 3.5.1:

Divide both sides by $5$ . $\frac{5 k}{5} = \frac{- 14}{5}$

Step 3.5.2:

Simplify the left side.

Step 3.5.2.1:

Cancel out the common factor of $5$ . $\frac{5 k}{5} = \frac{- 14}{5}$

Step 3.5.2.1.2:

Simplify the division by $1$ . $k = \frac{- 14}{5}$

Step 3.5.3:

Simplify the right side.

Step 3.5.3.1:

Write the negative sign in front of the fraction. $k = - \frac{14}{5}$

Step 4:

Present the solution in various forms.

Exact Form: $k = - \frac{14}{5}$

Decimal Form: $k = - 2.8$

Mixed Number Form: $k = - 2 \frac{4}{5}$

Knowledge Notes:

To solve an equation where the bases are powers of the same number, one can convert the bases to be identical and then equate the exponents. This is possible because, according to the properties of exponents, if $a^{m} = a^{n}$ for some non-zero base $a$ , then $m = n$ .

In this problem, we used the fact that $8 = 2^{3}$ and $4 = 2^{2}$ to rewrite both sides of the equation with a common base of 2. Once the bases are the same, we can set the exponents equal to each other and solve for the unknown variable $k$ .

The distributive property, which states that $a (b + c) = a b + a c$ , is applied to expand expressions such as $3 (3 k + 4)$ and $2 (2 k - 1)$ .

When solving linear equations, we aim to isolate the variable on one side of the equation. This often involves moving terms containing the variable to one side and constants to the other, and then simplifying by combining like terms and performing basic arithmetic operations (addition, subtraction, multiplication, and division).

Finally, the solution can be presented in various forms, including exact form (as a fraction), decimal form, and mixed number form, depending on the context or requirements of the problem.