Problem 12 - Algebra 2

Question

Solve. \(3+(4-x)^{\frac{3}{2}}=11\)

Step-by-Step Solution

Answer

The solution to the equation \(3+(4-x)^{\frac{3}{2}}=11\) is \(x = 0\).

Step 1: Isolate the term with the fractional exponent

Subtract 3 from both sides of the equation to isolate the term with the fractional exponent. This results in \((4-x)^{\frac{3}{2}} = 11 - 3 = 8\).

Step 2: Get rid of the fractional power

To remove the fractional power, raise both sides of the equation to the reciprocal of the fractional power, which is \(\frac{2}{3}\). This results in \( \sqrt[3]{(4-x)^2} = \sqrt[3]{(8)^2}\). Simplify this to yield \(4-x = 2\sqrt[3]{8}\).

Step 3: Simplify and solve for x

Simplify right side to get \(4 - x = 2(2) = 4\). Now, solve for x by subtracting 4 from each side to get \(-x = 0\) which in turn gives \(x = 0\).

Key Concepts

Fractional Exponents

Fractional exponents can seem intimidating at first glance, but breaking them down makes them easier to understand. When you see an exponent in the form \( a^{m/n} \), it essentially represents the use of roots and powers together. Here's how to visualize it:

• The denominator of the fraction \( n \) tells you the root, specifically the \( n \)-th root.
• The numerator \( m \) represents the power to which you'll raise your base after taking the root.

For example, \( (4-x)^{3/2} \) can be interpreted as taking the square root of \( 4-x \) and then cubing that result. Knowing this, the power of fractional exponents can be easily managed by following a careful sequence of operations in any algebraic expression.

Isolation of Terms

One central strategy in solving algebraic equations is the isolation of terms. This means rearranging the equation in such a way that one term is "alone" on one side of the equality. For the equation \( 3+(4-x)^{\frac{3}{2}}=11 \), begin by isolating \((4-x)^{\frac{3}{2}}\) on one side. You do this by subtracting 3 from both sides of the equation:

• The equation becomes \( (4-x)^{\frac{3}{2}} = 8 \).

This step simplifies the equation, making subsequent steps more manageable. With isolated terms, your next actions follow logically, to solve for the variable, ensuring your solution is streamlined and efficient.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying equations to isolate variables and solve them. After isolating a term, the next step often involves simplifying expressions further. Let's consider our exercise: once \((4-x)^{\frac{3}{2}}\) is isolated, raise both sides to the reciprocal of \(\frac{3}{2}\), which is \(\frac{2}{3}\). This eliminates the fractional exponent:

• Perform \( \sqrt[3]{(4-x)^2} = \sqrt[3]{8^2} \).
• The equation simplifies to \( 4-x = 2\sqrt[3]{8} \).

The cube root of 8 is 2, so the equation simplifies further to \( 4-x = 4 \). Finally, solve by subtracting 4 from each side, yielding \( -x = 0 \) and \( x = 0 \). Through careful manipulation, the solution becomes clear, illustrating the power of methodically working through algebraic equations.