Problem 91 - College Algebra Essentials

Question

Find all values of \(x\) satisfying the given conditions. $$y=(x+4)^{\frac{3}{2}} \text { and } y=8$$

Step-by-Step Solution

Answer

The value of \(x\) that satisfies the given conditions is \(0\).

Step 1: Substitute the value of \(y\) from the second equation into the first

First we make both equations equal by setting the right sides of each equation equal to each other since \(y\) is the same in both. Therefore, we have \( (x+4)^{\frac{3}{2}} = 8 \)

Step 2: Simplifying the equation

To cancel the cube root on the left side, we can raise both sides of the equation to the power of \(\frac{2}{3}\). Thus, the equation becomes: \( ((x+4)^{\frac{3}{2}})^{\frac{2}{3}} = 8^{\frac{2}{3}} \). Simplifying this gives \(x+4 = 2^2 = 4 \)

Step 3: Solve for \(x\)

Finally, rearranging the equation to isolate \(x\), we get: \(x = 4 - 4 = 0\)

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to a power, or exponent. When dealing with exponentiation, the base is multiplied by itself a specified number of times dictated by the exponent. For example, in the expression

• \((x+4)^{\frac{3}{2}}\), the base is \((x+4)\) and the exponent is \(\frac{3}{2}\)

This particular exponentiation is a fractional exponent, which combines the concepts of both roots and powers:

• The numerator \(3\) indicates cube (raising to the third power).
• The denominator \(2\) indicates taking the square root.

Thus, \((x+4)^{\frac{3}{2}}\) means taking the square root of \((x+4)\) and then cubing the result. Understanding how to work with fractional exponents is crucial when solving equations that involve these operations, as seen in this example where we simplify the expression to find \(x\).

Equations with Roots

Equations with roots involve expressions that include roots like square roots, cube roots, and in this case, the expression with a fractional exponent which also represents roots. Solving these equations often requires manipulating the expression to remove the root so we can simplify the equation effectively.
In the given problem, we tackled the equation

• \((x+4)^{\frac{3}{2}} = 8\)

To simplify this: raise both sides to the reciprocal of the root part. Here, that means raising both sides by \(\frac{2}{3}\). This operation neutralizes the \(\frac{3}{2}\) exponent on the left-hand side as follows:

• \(((x+4)^{\frac{3}{2}})^{\frac{2}{3}} = 8^{\frac{2}{3}}\)
• Simplifying gives \(x+4 = 4\)

Understanding equations with roots and how to handle them by utilizing laws of exponents and roots is essential for solving such problems.

Substitution Method

The substitution method is a technique used to solve systems of equations by replacing a variable with an equivalent expression. In this problem, we use it to derive a single equation with one variable by using known values.
Here, both equations share the common variable \(y\).
This allows us to substitute directly and simplify:

• Start with \(y=(x+4)^{\frac{3}{2}} \) and \(y=8\)
• Substitute \(y=8\) into the equation \((x+4)^{\frac{3}{2}} = 8\)

This substitution reduces the complexity of having multiple variables and results in an equation solely in terms of \(x\). This streamlines the solving process, leading us directly to the solution where \(x\) can be found only when the equation is simplified properly.