Problem 40 Solve each equation. $$ \sqr... [FREE SOLUTION]

Chapter 10: Problem 40

Solve each equation. $$ \sqrt[3]{r^{2}+2 r+8}=\sqrt[3]{r^{2}+3 r+12} $$

Short Answer

Expert verified

r = -4

Step by step solution

Equalize the Expressions

Since the cube roots on both sides are equal, set the radicands (the expressions inside the cube roots) equal to each other:\[ r^2 + 2r + 8 = r^2 + 3r + 12 \]

Simplify the Equation

Subtract $ r^2 $ from both sides of the equation to simplify:\[ 2r + 8 = 3r + 12 \]

Isolate the Variable

Subtract 2r from both sides to isolate the variable on one side:\[ 8 = r + 12 \]

Solve for r

Subtract 12 from both sides to solve for r:\[ 8 - 12 = r \]\[ r = -4 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

cube roots

To understand how to solve equations involving cube roots, it's crucial to know what cube roots are. A cube root of a number is a value that, when multiplied by itself three times, gives the original number. For instance, the cube root of 27 is 3 because $3 \times 3 \times 3 = 27$.
Cube roots are denoted using the radical symbol with a small 3 above it, such as \sqrt[3]{x}\. In the equation from the example, \sqrt[3]{r^{2}+2r+8}\ and \sqrt[3]{r^{2}+3r+12}\ are cube roots.
Solving cube root equations involves removing the cube root by cubing both sides of the equation. However, in the given exercise, both sides already have cube roots, simplifying the process.

radicands

Radicands are the numbers or expressions inside a radical symbol. For cube roots, the radicand is the expression under the cube root sign. In our problem, the radicands are $r^{2}+2r+8$ and $r^{2}+3r+12$.
Setting the radicands equal is a key step in solving our equation because \sqrt[3]{a} = \sqrt[3]{b}\ implies $a = b$.
So, we set $r^{2}+2r+8$ equal to $r^{2}+3r+12$ which simplifies our equation and eliminates the cube roots.

simplifying equations

Simplifying equations involves combining like terms and reducing expressions to their simplest form. Our example shows this process clearly.
After setting the radicands equal, we get $r^2 + 2r + 8 = r^2 + 3r + 12$. The next step is to simplify by eliminating like terms. Subtract $r^2$ from both sides:
$2r + 8 = 3r + 12$
This makes the equation simpler and easier to solve.

isolating variables

Isolating the variable means getting the variable on one side of the equation by itself. In our simplified equation $2r + 8 = 3r + 12$, we need to isolate r.
First, subtract 2r from both sides:
$8 = r + 12$
Then, subtract 12 from both sides to solve for r:
$8 - 12 = r$
Simplifying this gives \r = -4\.

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91影视

Solve each equation. $$ \sqrt[3]{r^{2}+2 r+8}=\sqrt[3]{r^{2}+3 r+12} $$

Short Answer

Step by step solution

Equalize the Expressions

Simplify the Equation

Isolate the Variable

Solve for r

Key Concepts

cube roots

radicands

simplifying equations

isolating variables

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