Question

Solve the equation. \(512^{5 x-1}=\left(\frac{1}{8}\right)^{-4-x}\)

Short answer

The solution for \(x\) is 0.5

Step-by-step solution

Simplify bases

The base on the left side is 512, while the base on the right side is 1/8. Since these bases are not clearly related, we should try to rewrite them in terms of the same base. Luckily, both 512 and 1/8 can be written in terms of 2. \(512 = 2^9\) and \(\frac {1}{8} = 2^{-3}\). Thus, the equation becomes: \[(2^9)^{5x-1} = (2^{-3})^{-4-x}\] .

Simplify exponents

Use the law of exponents that says \((a^b)^c=a^{bc}\). This can be used to simplify the equation further: \[ 2^{9*(5x-1)} = 2^{-3*(-4-x)}\] Simplify this to get: \[ 2^{45x-9} = 2^{12+3x}\].

Set exponents equal to each other

Since the bases are equal (both are 2), we can equate the exponents: \(45x - 9 = 12 + 3x\).

Solve linear equation

Solving for \(x\), we first isolate items that contain \(x\) on one side and the rest on the other side, resulting in: \(45x - 3x = 12 + 9\). This simplifies down to: \(42x = 21\). By dividing both sides by 42, we find that \(x = 21/42 = 0.5\).

Key concepts

Laws of Exponents

Understanding the laws of exponents is crucial when working with exponential equations. The laws of exponents, also known as the rules of exponents, are a set of rules that describe how to handle various operations involving exponents. Here are some of the basic laws:

• \textbf{Product of Powers:} For any nonzero number a, and any integers m and n, the rule is \(a^m \times a^n = a^{m+n}\).
• \textbf{Power of a Power:} For any nonzero number a, and any integers m and n, the rule is \( (a^m)^n = a^{mn}\).
• \textbf{Power of a Product:} For the product of any nonzero numbers a and b, and any integer n, the rule is \( (ab)^n = a^n \times b^n\).
• \textbf{Negative Exponent:} For any nonzero number a, the rule is \( a^{-n} = \frac{1}{a^n}\), which implies that negative exponents represent the reciprocal of the base raised to the opposite positive power.
• \textbf{Zero Exponent:} For any nonzero number a, the rule is \(a^0 = 1\).

Applying these rules correctly is essential for simplifying exponents and solving related equations. In our example with the equation \(512^{5x-1} = (\frac{1}{8})^{-4-x}\), we needed to rewrite both sides of the equation using a common base, which was base 2. Then, using the rule of 'Power of a Power', we were able to simplify the equation further, making it easier to solve.

Simplify Exponents

To simplify exponents, we often need to apply one or more laws of exponents. When the bases are the same and the terms are multiplied, you add the exponents. When an exponent is raised to another exponent, you multiply them.

For instance, with our problem, we rewrote 512 as \(2^9\) and \(\frac{1}{8}\) as \(2^{-3}\). We then simplified the equation using the 'Power of a Power' law to combine the exponents on each side. A key point here is that simplification often makes it much easier to see the relationship between the two sides of an equation. For students, it's important to remember that the step of simplifying an exponential expression can drastically reduce the complexity of the problem and this step should be done carefully to avoid mistakes.

Linear Equation Solving

Once the exponents are simplified, we often end up with a linear equation, which is an equation of the first degree, meaning it has one variable with an exponent of one. The standard form of a linear equation is \(Ax + B = 0\), where A and B are constants, and x is the variable we solve for.

Applying this to our example, after simplifying the exponents, we used the basic principles of linear equation solving. We collected like terms on each side, which means getting all the terms with the variable on one side and the constants on the other. Then we isolated the variable by dividing through by the coefficient, resulting in \(x = 21/42\).

Tips for Solving Linear Equations:

• Collect like terms on both sides of the equation.
• Move all terms containing variables to one side of the equation.
• Combine like terms to simplify.
• Isolate the variable by performing inverse operations.
• Check your answer by plugging it back into the original equation.

In our case, isolating \(x\) allowed us to find that \(x = 0.5\), completing the solution to the equation.