Problem 21 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation by hand. Do not use a calculator. $$x^{2 / 5}=4$$

Step-by-Step Solution

Answer

The solutions are \(x = 32\) and \(x = -32\).

Step 1: Rewrite the equation using exponents

To isolate the variable, identify that the exponent is with a base of 5. Our equation is currently in the form \(x^{2/5} = 4\). To simplify solving for \(x\), rewrite the equation as \((x^{1/5})^2 = 4\).

Step 2: Apply the inverse operation to the exponent

To remove the power of 2, take the square root on both sides of the equation: \((x^{1/5}) = \pm\sqrt{4}\). Simplifying the square root gives us \(x^{1/5} = \pm2\).

Step 3: Solve for x by expanding the base

Now, to clear the \(1/5\) power from \(x\), we raise both sides of the equation to the power of 5. This operation gives \(x = (\pm2)^5\).

Step 4: Calculate the solution

Compute \((2)^5 = 32\) and \((-2)^5 = -32\). Therefore, the solutions for \(x\) are \(x = 32\) and \(x = -32\).

Key Concepts

Exponents

Exponents are mathematical notations used to express repeated multiplication of a number by itself. An exponent is written as a superscript, and it tells us how many times the base is multiplied. For example, in the term \(x^{2/5} \), "2/5" is the exponent, and "x" is the base. Here, exponents can represent fractional values making them useful for denoting roots combined with powers.

• Understand that \(x^{2/5} \) means \((x^{1/5})^2 \). The part "\(x^{1/5}\)" is the fifth root of "x" raised to the power of 2.
• When dealing with fractional exponents, the numerator signifies the power, while the denominator indicates the root.

By getting comfortable with exponents, you can simplify complex equations and solve them more easily.

Inverse Operations

Inverse operations are operations that reverse the effects of each other. When solving equations, we use inverse operations to isolate the variable.

• For example, if an exponent operation is applied to a variable, its inverse is typically a root. Taking the square root of both sides reverses squaring, helping to solve for the unknown variable.
• In the given problem, after identifying the equation \((x^{1/5})^2 = 4\), to counter the square (the power of 2), the square root was applied: \(x^{1/5} = \pm\sqrt{4} \).

These operations help to break down and solve equations by transforming one side at a time to balance the equation.

Powers

Powers are closely related to exponents and describe the repeated multiplication of a number. When you raise a number to a power, you multiply it by itself as many times as specified by the power.

• For example, \((\pm2)^5\) means you multiply "\(\pm2\)" by itself five times. The power, which is the small raised number, tells us how many times to multiply the base with itself.
• This process was utilized in the exercise to calculate \((\pm2)^5\) to finalize the solution for \(x\).

Understanding how powers work is essential in algebra where combining and simplifying expressions often involves manipulating powers.

Square Root

The square root is a special number which, when multiplied by itself, gives the original number. It's symbolized by the radical sign "√".

• In the problem, when we encounter \(x^{1/5} = \pm\sqrt{4}\), it means to find a number which multiplied by itself results in 4. The square roots of 4 are 2 and -2 because \(2^2 = 4\) and \((-2)^2 = 4\).
• Taking a square root is an important tool in reversing a square in algebra. It helps in simplifying and solving equations effectively.

Recognizing how to apply square roots allows you to solve quadratic equations by returning the equation to its original form.