Question

Find the real solution(s) of the equation. Round your answer to two decimal places when appropriate. (See Example 4.) \(7 x^4=56\)

Short answer

The real value of \(x\) that solves the equation \(7 x^4 = 56\) is \(x = 1.68\).

Step-by-step solution

Simplify the Equation

First, the given equation, \(7 x^4 = 56\) can be simplified by dividing both sides of the equation by 7. This will leave \(x^4 = 8\).

Take the Fourth Root

To solve for \(x\), it is necessary to take the fourth root of both sides of the equation. This results in \(x = \sqrt[4]{8}\).

Calculate the Value of x

Finally, calculate the fourth root of 8. Rounding to two decimal places, this results in a real value for \(x = 1.68\).

Key concepts

Equation Solving

Algebra is all about finding the values of variables that make an equation true. Solving an equation means we want to isolate the variable of interest by performing operations like addition, subtraction, multiplication, and division. In the exercise given, we start with an equation: \( 7 x^4 = 56 \). The goal is to solve for \( x \).

• First, simplify the equation: divide both sides by 7 to get \( x^4 = 8 \).
• Next, to solve for \( x \), notice the power of 4. This means we will take the fourth root on both sides.

By simplifying step-by-step, you isolate the variable \( x \), finding the value that satisfies the equation.

Fourth Roots

Finding the fourth root is a specific operation in algebra. It is closely related to squaring and taking square roots, but instead involves fourfold multiplication or division.

• The fourth root of a positive number \( a \) is the number \( b \) such that \( b^4 = a \).
• In our exercise, we need \( x = \sqrt[4]{8} \).

Calculating \( \sqrt[4]{a} \) generally involves finding the value that, when used as a factor four times, results in the original number. Computations can be efficiently performed using calculators or software that supports root functions.

Rounding Decimals

In mathematics, results don't always come out as neat, whole numbers. Often, we have to round numbers to make them easier to use, especially when solutions are needed in decimal forms.

• Rounding involves looking at the digit right after the place you want to round to. If it's 5 or greater, you round up.
• In this exercise, when we calculate the fourth root of 8, we get 1.6817928305... and are asked to round to two decimal places.
• This means looking at the third decimal place. Since it's an 8, we round the second decimal place from 1.68 to 1.68.

Rounding simplifies numbers so they are easier to interpret and use, especially in practical applications like science and engineering.