Solving Functions: $y=x^{n}$ And Given Y Values

Hey guys! Today, we're diving into the exciting world of functions, specifically those in the form of $y = x^n$, where we're given a value for y and need to figure out what x is. It might sound a bit intimidating at first, but trust me, it’s super manageable once you understand the basics. We'll break down four different problems step by step, so you can see exactly how it's done. So, grab your thinking caps, and let’s get started!

1) Finding x when $y = x^{\frac{4}{3}}$ and $y = 625$

Let's kick things off with our first problem: $y = x^{\frac{4}{3}}$ and $y = 625$. Our mission is to find the value of x that makes this equation true. The key here is understanding fractional exponents. Remember, a fractional exponent like $\frac{4}{3}$ means we're dealing with both a power and a root. Specifically, $x^{\frac{4}{3}}$ can be thought of as the cube root of x raised to the fourth power, or $(\sqrt[3]{x})^4$.

So, how do we tackle this? Well, we need to isolate x. To do that, we'll use the inverse operation of raising to the power of $\frac{4}{3}$, which is raising to the power of $\frac{3}{4}$. Think of it like this: if we have a fraction as an exponent, we flip the fraction and use that as our new exponent to undo the original one. This is because $(x^{\frac{4}{3}})^{\frac{3}{4}} = x^{(\frac{4}{3} * \frac{3}{4})} = x^1 = x$. Cool, right?

Now, let's apply this to our equation. We have $625 = x^{\frac{4}{3}}$. We'll raise both sides of the equation to the power of $\frac{3}{4}$: $(625)^{\frac{3}{4}} = (x^{\frac{4}{3}})^{\frac{3}{4}}$. This simplifies to $(625)^{\frac{3}{4}} = x$. So, now we just need to figure out what $(625)^{\frac{3}{4}}$ is. Remember, this means we're taking the fourth root of 625 and then cubing the result. The fourth root of 625 is 5 (since $5*5*5*5 = 625$). Then, we cube 5: $5^3 = 5 * 5 * 5 = 125$. Therefore, $x = 125$.

To double-check our work, we can plug this value of x back into our original equation: $y = (125)^{\frac{4}{3}}$. The cube root of 125 is 5, and $5^4 = 625$. Bingo! It works. So, for the first function, the value of x is 125 when y is 625.

2) Finding x when $y = x^{\frac{6}{5}}$ and $y = 64$

Alright, let's move on to the second problem: $y = x^{\frac{6}{5}}$ and $y = 64$. Just like before, our goal is to isolate x. This time, we have a fractional exponent of $\frac{6}{5}$. So, we're going to raise both sides of the equation to the power of $\frac{5}{6}$. Remember, we flip the fraction to get the inverse operation.

Our equation is $64 = x^{\frac{6}{5}}$. Raising both sides to the power of $\frac{5}{6}$, we get: $(64)^{\frac{5}{6}} = (x^{\frac{6}{5}})^{\frac{5}{6}}$. This simplifies to $(64)^{\frac{5}{6}} = x$. So, we need to calculate $(64)^{\frac{5}{6}}$. This means we take the sixth root of 64 and then raise the result to the fifth power. The sixth root of 64 is 2 (since $2*2*2*2*2*2 = 64$). Now, we raise 2 to the fifth power: $2^5 = 2 * 2 * 2 * 2 * 2 = 32$. Therefore, $x = 32$.

Let’s verify our solution. We plug x = 32 back into the original equation: $y = (32)^{\frac{6}{5}}$. The fifth root of 32 is 2, and $2^6 = 64$. Awesome! It checks out. So, for the second function, x is 32 when y is 64.

3) Finding x when $y = x^{\frac{3}{2}}$ and $y = 216$

Now, let's tackle the third function: $y = x^{\frac{3}{2}}$ and $y = 216$. Same game plan, guys! We need to isolate x. We have a fractional exponent of $\frac{3}{2}$, so we'll raise both sides of the equation to the power of $\frac{2}{3}$.

Our equation is $216 = x^{\frac{3}{2}}$. Raising both sides to the power of $\frac{2}{3}$, we get: $(216)^{\frac{2}{3}} = (x^{\frac{3}{2}})^{\frac{2}{3}}$. This simplifies to $(216)^{\frac{2}{3}} = x$. Now we need to calculate $(216)^{\frac{2}{3}}$. This means we take the cube root of 216 and then square the result. The cube root of 216 is 6 (since $6*6*6 = 216$). Then, we square 6: $6^2 = 6 * 6 = 36$. Therefore, $x = 36$.

Let's make sure we’re right. We plug x = 36 back into the original equation: $y = (36)^{\frac{3}{2}}$. The square root of 36 is 6, and $6^3 = 216$. Perfect! It works. So, for the third function, x is 36 when y is 216.

4) Finding x when $y = x^{\frac{7}{3}}$ and $y = 128$

Last but not least, let's tackle the fourth function: $y = x^{\frac{7}{3}}$ and $y = 128$. You know the drill by now! We need to isolate x. We have a fractional exponent of $\frac{7}{3}$, so we’ll raise both sides of the equation to the power of $\frac{3}{7}$.

Our equation is $128 = x^{\frac{7}{3}}$. Raising both sides to the power of $\frac{3}{7}$, we get: $(128)^{\frac{3}{7}} = (x^{\frac{7}{3}})^{\frac{3}{7}}$. This simplifies to $(128)^{\frac{3}{7}} = x$. So, we need to calculate $(128)^{\frac{3}{7}}$. This means we take the seventh root of 128 and then cube the result. The seventh root of 128 is 2 (since $2^7 = 128$). Then, we cube 2: $2^3 = 2 * 2 * 2 = 8$. Therefore, $x = 8$.

To be absolutely sure, let's plug x = 8 back into the original equation: $y = (8)^{\frac{7}{3}}$. The cube root of 8 is 2, and $2^7 = 128$. Yes! It checks out. So, for the fourth function, x is 8 when y is 128.

Summary and Key Takeaways

So, there you have it! We've successfully solved for x in four different functions of the form $y = x^n$. The main takeaway here is understanding how to deal with fractional exponents. Remember, a fractional exponent represents both a root and a power. To isolate x, we raise both sides of the equation to the reciprocal of the fractional exponent. This effectively “undoes” the original exponent and allows us to solve for x.

• For $y = x^{\frac{4}{3}}$ and $y = 625$, we found $x = 125$.
• For $y = x^{\frac{6}{5}}$ and $y = 64$, we found $x = 32$.
• For $y = x^{\frac{3}{2}}$ and $y = 216$, we found $x = 36$.
• For $y = x^{\frac{7}{3}}$ and $y = 128$, we found $x = 8$.

Keep practicing, guys, and you'll become pros at solving these types of problems in no time! Understanding exponents and roots is crucial in math, and this is a fantastic step in that direction. If you ever get stuck, just remember to flip the fraction and raise to that power – you got this! 💯