Solving Cube Root Equations A Step-by-Step Guide

Hey there, math enthusiasts! Ever stumbled upon a cube root equation and felt a little lost? Don't worry, you're not alone! Cube root equations can seem intimidating at first, but with a little guidance, they're totally conquerable. In this article, we're going to break down the equation $\sqrt[3]{2x+5} = 5$ step-by-step, so you can not only solve it but also understand the underlying principles. Let's dive in and unravel this mathematical puzzle together!

Decoding the Cube Root Equation

So, what is the solution to the equation $\sqrt[3]{2x+5} = 5$? This might look like a jumble of symbols and numbers, but let's break it down. The core of this equation is the cube root, represented by the radical symbol with a small '3' nestled in its crook. Remember, a cube root is the inverse operation of cubing a number. Just like the square root asks, "What number multiplied by itself equals this?", the cube root asks, "What number multiplied by itself twice equals this?".

In our equation, we're tasked with finding the value of 'x' that makes the equation true. Think of it like a treasure hunt where 'x' is the hidden treasure, and the equation is our map. The equation tells us that if we take twice 'x', add 5 to it, and then find the cube root of the result, we should end up with 5. Our mission, should we choose to accept it (and we do!), is to reverse these operations and find 'x'.

Before we start crunching numbers, let's pause and appreciate the elegance of this equation. It's a neat little package of mathematical concepts, combining algebra and the idea of inverse operations. Understanding these concepts is crucial, not just for solving this particular equation, but for tackling a whole range of mathematical challenges. So, let's put on our thinking caps and get ready to embark on this mathematical adventure!

Step 1: Eliminating the Cube Root

The first order of business in solving our equation, $\sqrt[3]{2x+5} = 5$, is to eliminate the cube root. Remember, the cube root is like a mathematical lockbox, and we need the right key to open it. That key, my friends, is cubing! Cubing is the inverse operation of taking the cube root, meaning it undoes the cube root's effect. It's like having a mathematical antidote.

To eliminate the cube root, we need to cube both sides of the equation. This is a crucial step because it maintains the balance of the equation. Think of an equation like a seesaw – if you do something to one side, you have to do the same to the other side to keep it level. So, we'll raise both sides of the equation to the power of 3.

On the left side, we have $(\sqrt[3]{2x+5})^3$. When we cube a cube root, the operations cancel each other out, leaving us with just the expression inside the cube root, which is $2x + 5$. It's like the lockbox has been opened, and we can finally see what's inside!

On the right side, we have $5^3$, which means 5 multiplied by itself twice: $5 * 5 * 5$. This gives us 125. So, our equation now looks much simpler: $2x + 5 = 125$. We've successfully banished the cube root and transformed our equation into a more manageable form. It's like we've cleared the first hurdle in our mathematical race!

Step 2: Isolating the Variable

Now that we've eliminated the cube root, our equation looks much friendlier: $2x + 5 = 125$. The next step in our mathematical treasure hunt is to isolate the variable 'x'. This means we want to get 'x' all by itself on one side of the equation, like giving it its own little island. To do this, we need to undo any operations that are being performed on 'x', one step at a time.

Currently, 'x' is being multiplied by 2 and has 5 added to it. Remember the order of operations (PEMDAS/BODMAS)? We need to reverse that order when isolating the variable. So, we'll deal with the addition first. To undo adding 5, we subtract 5 from both sides of the equation. Again, we're maintaining the balance of our seesaw! This gives us:

$2x + 5 - 5 = 125 - 5$

Simplifying this, we get:

$2x = 120$

Great! We've managed to peel away the addition and bring 'x' closer to isolation. Now, we just have one more operation to undo: the multiplication by 2. It's like we're slowly unwrapping a gift, revealing the value of 'x' piece by piece.

Step 3: Solving for 'x'

We're in the home stretch now! Our equation is currently $2x = 120$. We've isolated the variable as much as possible, and the only thing standing between us and the solution is that pesky 2 multiplying 'x'. To get 'x' completely alone, we need to undo this multiplication. And how do we undo multiplication? With division, of course!

To get rid of the 2, we'll divide both sides of the equation by 2. Remember, maintaining balance is key! This gives us:

$\frac{2x}{2} = \frac{120}{2}$

On the left side, the 2s cancel each other out, leaving us with just 'x'. On the right side, 120 divided by 2 is 60. So, our equation simplifies to:

$x = 60$

Eureka! We've found our treasure! The value of 'x' that satisfies the equation $\sqrt[3]{2x+5} = 5$ is 60. It's like we've reached the end of our mathematical quest and claimed our prize. But our journey doesn't end here. It's always a good idea to check our answer to make sure it's correct.

Step 4: Verifying the Solution

We've arrived at a potential solution, $x = 60$, but before we declare victory, it's crucial to verify the solution. This is like double-checking your map to make sure you've actually reached the treasure and haven't just stumbled into a mirage. To verify our solution, we'll substitute $x = 60$ back into the original equation and see if it holds true.

Our original equation was $\sqrt[3]{2x+5} = 5$. Let's plug in $x = 60$:

$\sqrt[3]{2(60)+5} = 5$

Now, we simplify the expression inside the cube root:

$\sqrt[3]{120+5} = 5$

$\sqrt[3]{125} = 5$

And finally, we ask ourselves, what is the cube root of 125? Well, 5 multiplied by itself three times (5 * 5 * 5) equals 125. So, the cube root of 125 is indeed 5!

$5 = 5$

Our equation holds true! This confirms that our solution, $x = 60$, is correct. We've not only found the treasure but also verified its authenticity. It's like we've completed our mathematical mission with flying colors!

The Final Answer and Key Takeaways

So, what is the solution to the equation $\sqrt[3]{2x+5} = 5$? The answer, my friends, is C. 60! We've successfully navigated the twists and turns of this cube root equation and emerged victorious. But more importantly, we've learned some valuable skills along the way.

We've seen how to eliminate cube roots by cubing both sides of the equation, how to isolate the variable by undoing operations in the reverse order, and how to verify our solution to ensure its accuracy. These are powerful tools that you can use to tackle a wide range of algebraic equations. So, the next time you encounter a cube root equation, don't shy away from the challenge. Remember the steps we've learned, and you'll be well on your way to unlocking the solution!

Keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this!