Question

Find the real solution(s) of the equation. Round your answer to two decimal places when appropriate. (See Example 4.) \((x-5)^4=256\)

Short answer

There are two real solutions to the equation \((x-5)^4=256\), which are \(x=9.00\) and \(x=1.00\).

Step-by-step solution

Rewrite the given equation

The equation given is \((x-5)^4=256\). We need to solve this equation for x.

Take the fourth root of both sides

The fourth root of both sides must be taken in order to make the base and exponent on the left side of the equation equal to 1. The fourth root of 256 is 4 and -4 (since any even root of a number also has a negative solution). The equation now becomes \(x-5=4\) and \(x-5=-4\).

Solve the two equations for x

The two equations \(x-5=4\) and \(x-5=-4\) can be solved individually for x. Solving \(x-5=4\) gives \(x=9\) and solving \(x-5=-4\) gives \(x=1\).

Key concepts

Understanding Fourth Roots

When you see an equation like \((x-5)^4=256\), you're dealing with something raised to the fourth power. To break this down, you need to think about the opposite operation, which is taking the fourth root. Taking the fourth root of a number means finding a value that, when raised to the power of four, gives you the original number. For example, the fourth root of 256 is 4 because \(4^4 = 256\). This means

• \(4 \times 4 \times 4 \times 4 = 256\)

One important aspect of fourth roots—and any even roots, like square roots—is that they have both positive and negative solutions. So, when we say the fourth root of 256, we mean both 4 and -4, because

• \((-4)^4 = 256\)

This dual result requires us to write two equations: one for the positive fourth root and one for the negative. This is how we extend our solution path.

Exponents and Powers

Exponents, like the 4 in \((x-5)^4=256\), tell us how many times we multiply a base number by itself. Let's break down what this means: In the expression \((x-5)^4\), "x-5" is the base, and the 4 is the exponent.

• The base is the number or expression getting multiplied.
• The exponent (or power) tells you how many times you multiply "x-5" by itself.

The process of finding what to multiply by itself is called exponentiation. In this case, the equation becomes simpler when you take the fourth root of both the sides as we discussed earlier. This reduces it to two separate linear equations to solve for \(x\). Exponential operations are essential in determining roots and making calculations easier, especially when simplifying complex equations.

Steps in Solving Equations

The given equation \((x-5)^4=256\) involves solving for \(x\) through a series of logical steps.First, we simplify the expression. We take the fourth root of both sides, just as we would take the square root if the exponent were 2. By doing this,

• We break it down into \(x-5 = 4\) and \(x-5 = -4\).

Finally, solving these two linear equations will provide the solutions for \(x\). Let's do that:

• If \(x-5=4\), then \(x=9\).
• If \(x-5=-4\), then \(x=1\).

By following these steps methodically, you arrive at the solutions \(x=9\) and \(x=1\). Understanding this process demonstrates the utility of breaking problems down into smaller, manageable parts and applying inverse operations like taking roots to find solutions conveniently.