Question

In Exercises 15–26, solve the equation. Check your solution(s). $$ \sqrt[3]{2 x-5}-\sqrt[3]{8 x+1}=0 $$

Short answer

The solution for the equation \(\sqrt[3]{2 x-5}-\sqrt[3]{8 x+1}=0\) is \(x = -\frac{2}{3}\).

Step-by-step solution

Rearrangement to isolate a cube root

Firstly, rearrange the given equation to isolate the cube root terms by moving \(\sqrt[3]{8x+1}\) to the other side of the equality. This gives you: \(\sqrt[3]{2x-5} = \sqrt[3]{8x+1}\).

Cubing both sides

To get rid of the cube roots, raise both sides of the equation to the power three: \((\sqrt[3]{2x-5})^3 = (\sqrt[3]{8x+1})^3\). This simplification leads to the equation \(2x-5 = 8x+1\).

Simplify the equation to solve for x

Rearrange the equation by subtracting \(2x\) from both sides and subtracting 1 from both sides to isolate \(x\):\(2x - 2x - 5 + 1 = 8x - 2x + 1 - 1\) will give you \(x = -\frac{2}{3}\).

Check the solution

Finally, plug in the calculated \(x\) into the original equation to verify correctness: \(\sqrt[3]{2(-\frac{2}{3})-5}-\sqrt[3]{8(-\frac{2}{3})+1}\) should be equal to 0. Simplify both cube root terms separately, if indeed they equate to 0 then the solution is correct.

Key concepts

Cube Roots

Cube roots are fascinating elements in mathematics that allow us to find a number which, when multiplied by itself three times, gives the original number. For example, the cube root of 27 is 3, since multiplying 3 by itself three times (3 × 3 × 3) equals 27.
In notation, the cube root is often represented as \( \sqrt[3]{x} \). This symbol indicates that we are looking for a number which, cubed, will return \( x \). Cube roots are particularly useful when needing to simplify expressions involving cubic relationships or to reverse the process of cubing a number.
In equations, cube roots are often isolated to make them easier to work with, as seen in the exercise where terms were rearranged to isolate each cube root. Learning how to effectively manipulate cube roots is a key skill that helps when solving complex algebraic equations.

Solution Verification

Solution verification is a crucial step in solving any equation. It helps ensure that the answer derived actually satisfies the original equation. In our exercise, after finding a potential solution for \( x \), we substituted it back into the original equation. This step checks whether both sides of the equation balance, confirming the correctness of the solution.
If substituting back the value results in a true statement, then the solution is verified. If not, we must re-evaluate our steps or consider other potential solutions.
Verification is important not just to avoid errors, but because some manipulations of equations, like squaring both sides, can introduce extraneous solutions which do not satisfy the original equation.

Algebraic Manipulation

Algebraic manipulation involves using various algebra rules to rearrange and simplify equations to find solutions. It’s all about using operations like addition, subtraction, multiplication, and division effectively and logically.
In our exercise, algebraic manipulation helped to rearrange the terms so that the cube roots could be addressed more easily. Once isolated, the equation was simplified to solve for \( x \).

• First, we moved terms involving \( x \) to one side to isolate \( x \).
• By subtracting \( 2x \) from both sides, we simplified the equation, reducing it to a solvable form.

Algebraic manipulation is a fundamental skill needed in all areas of mathematics to craft solutions neatly and efficiently.

Cubing Both Sides

Cubing both sides of an equation is a technique used to eliminate cube roots. It's the reverse operation of taking a cube root, allowing us to simplify equations significantly.
In the given problem, once cube roots were isolated, both sides of the equation were cubed. This process helps in transforming the cube root terms into linear terms, which are generally easier to solve.

• After cubing \( \sqrt[3]{2x-5} = \sqrt[3]{8x+1} \), we obtained the linear equation \( 2x-5 = 8x+1 \).
• This approach streamlined the path to finding \( x \).

While cubing, it's important to apply the operation symmetrically to prevent introducing errors or altering the equality of the equation. Understanding cubing both sides allows for more adaptable approaches to solving cubic equations.