Problem 31 - Algebra and Trigonometry

Question

Solve each equation with rational exponents in Exercises \(31-40\) Check all proposed solutions. $$x^{\frac{3}{2}}=8$$

Step-by-Step Solution

Answer

The solution to the equation \(x^{\frac{3}{2}} = 8\) is \(x = 4\).

Step 1: Understanding Rational Exponents

First, recognize that \(x^{a/b}\) is the same as \(\sqrt[b]{x^a}\). So in this case, \(x^{3/2}\) is equivalent to \(\sqrt{x^3}\). However, in order to simplify the process of solving, it will be more convenient to keep the original rational exponent form.

Step 2: Removing the Exponent

To remove an exponent of \(3/2\), we raise both sides of the equation to the reciprocal of \(3/2\) which is \(2/3\). Therefore, we have \((x^{3/2})^{2/3} = 8^{2/3}\). Simplifying \((x^{3/2})^{2/3}\) using the power of a power rule (where you multiply the exponents) results in \(x^1\) or simply \(x\).

Step 3: Solve for x

Compute \(8^{2/3}\), which is done by taking the cube root of \(8\) which is \(2\), and then squaring that result to get \(4\). This leads to \(x = 4\).

Step 4: Verification

To validate this result, substitute \(x = 4\) into the original equation. This gives us \((4)^{3/2} = 8\), which simplifies to \(8 = 8\), confirming that \(x = 4\) is indeed the correct solution.

Key Concepts

Rational Exponents Equations

Rational exponents are a way to express roots and powers together in a single expression. If you see an equation like \(x^{a/b} = c\), it's a rational exponent equation. The \(a/b\) is the rational exponent, where "a" is the power and "b" is the root.
In the equation \(x^{3/2} = 8\), \(3/2\) represents that the expression involves both squaring and rooting.

• The numerator "3" tells us to cube \(x\).
• The denominator "2" tells us to take the square root.

To solve these equations, balance the expression by applying the reciprocal of the exponent. This will simplify the equation and isolate the variable \(x\). By raising both sides to the reciprocal of \(3/2\), which is \(2/3\), you transform \(x^{3/2}\) into \(x^1\). The key takeaway is that rational exponents give us a clever way to work with roots and powers, streamlining complex operations.

Fractional Exponents

Fractional exponents like \(x^{3/2}\) appear frequently in algebra and mathematics in general, blending roots and powers.
They can make expressions easier to manage, especially when solving equations.
Understanding fractional exponents is crucial for simplifying expressions and solving equations effectively.Here's what they represent:

• The numerator of a fractional exponent represents the power, meaning how many times the base \(x\) is multiplied by itself.
• The denominator represents the root, showing the degree of the root — in this case, a square root because it's \(2\).

When you encounter \(x^{a/b}\), remember it can be rewritten as \[x^{a/b} = \left(\sqrt[b]{x}\right)^{a}\]This means you first perform the root operation and then apply the power, or vice versa, depending on which simplifies your work best. In practice, raising an expression with a fractional exponent to a power simplifies it significantly, such as \(x^{(3/2) \cdot (2/3)} = x^{1}\), which becomes just \(x\). Always focus on breaking down these compound operations.

Solving Algebraic Equations

Algebraic equations involve finding the value(s) of variables that satisfy the equation. The process requires a mix of understanding number operations and solving techniques.
With rational or fractional exponents, this involves:

• Identifying the structure of the equation and breaking it down into manageable parts.
• Using exponent rules, such as raising both sides to a reciprocal exponent to eliminate fractional exponents.

For example, to solve \(x^{3/2} = 8\), raise both sides by \(2/3\). This removes the exponent attached to \(x\):\[(x^{3/2})^{2/3} = 8^{2/3}\]This seems complex until broken down:

• Calculate \(8^{1/3}\), which is the cube root of \(8\) giving \(2\).
• Then square \(2\), resulting in \(4\).Now your \(x\) is isolated, giving \(x = 4\).

Verifying your solution is essential in algebra. Substitute \(x = 4\) back into the original equation to ensure both sides equal, confirming the solution is correct. This approach to solving involves dissecting the equation logically, maintaining balance, and verifying results.