Problem 11 - Algebra 2 and Trigonometry

Question

In \(3-17\) solve each equation and check. $$ 5 x^{\frac{3}{4}}=40 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 16\).

Step 1: Isolate the Variable Expression

To solve the equation \(5x^{\frac{3}{4}} = 40\), the first step is to isolate the expression involving \(x\). We do this by dividing both sides of the equation by 5, resulting in \(x^{\frac{3}{4}} = 8\).

Step 2: Eliminate the Exponent

To solve for \(x\), we need to eliminate the fractional exponent. Raise both sides of the equation to the power of \(\frac{4}{3}\) to cancel out the exponent on \(x\). This gives us \((x^{\frac{3}{4}})^{\frac{4}{3}} = 8^{\frac{4}{3}}\), which simplifies to \(x = 8^{\frac{4}{3}}\).

Step 3: Evaluate the Powers

We need to evaluate \(8^{\frac{4}{3}}\). First, compute the cube root of 8, which is 2. Then raise 2 to the power of 4: \(2^4 = 16\). Thus, \(8^{\frac{4}{3}} = 16\), so \(x = 16\).

Step 4: Check the Solution

To check our solution, substitute \(x = 16\) back into the original equation. We have \(5(16^{\frac{3}{4}}) = 40\). Compute \(16^{\frac{3}{4}}\): the fourth root of 16 is 2, and \(2^3 = 8\). Therefore, \(5 \times 8 = 40\), which is correct. Hence, our solution is verified.

Key Concepts

Rational Exponents

Rational exponents are a different way to express roots and powers in mathematics. When you see an exponent expressed as a fraction, such as \( x^{\frac{3}{4}} \), it's essentially indicating both a root and a power operation.

The numerator of the fraction (3 in this case) is the power, while the denominator (4) is the root. Therefore, \( x^{\frac{3}{4}} \) can be understood as the fourth root of \( x^3 \).

• Fractional exponents make it easier to handle expressions involving roots and powers, especially when solving equations.
• It's often beneficial to convert between radical form and fractional exponents to simplify calculations.

By using rational exponents, we can perform operations, like raising both sides of an equation to a power, more seamlessly as seen in our solution process. Understanding and manipulating these expressions is crucial for solving equations involving any form of powers or roots.

Isolation of Variable

When solving equations, one of the primary goals is to isolate the variable. This means rearranging the equation so that the variable of interest is by itself on one side of the equation.

In the given problem, \( 5x^{\frac{3}{4}} = 40 \), the first step was to divide both sides by 5. This isolates \( x^{\frac{3}{4}} \) on one side:

• This simplification yields \( x^{\frac{3}{4}} = 8 \).
• Once isolated, you can apply operations to solve for \( x \).

Always perform the same operation on both sides of the equation to maintain the equality.

Isolation is a critical skill as it simplifies complex equations and allows you to effectively use inverse operations to solve for the variable.

Checking Solutions

Once you have found a solution to an equation, it is vital to check that this solution satisfies the original equation.
After solving, we found that \( x = 16 \). Plugging it back into the original equation ensures our solution is correct:

• Calculate \( 16^{\frac{3}{4}} \): the fourth root of 16 is 2; then \( 2^3 = 8 \).
• Substitute into the equation: \( 5 \times 8 = 40 \).
• The left-side matches the right-side, confirming our solution's accuracy.

Checking your work is an essential step in solving equations. It verifies calculations and ensures the method used is correct. This practice helps build confidence and guarantees that no mistakes were made along the way.