Simplify.(y45)3\left(\frac{y^4}{5}\right)^3

y12125\frac{$$y^{12}$$}{125}125y12

Simplify. ( y 4 5 ) 3 \left(\frac{y^4}{5}\right)^3

So, let's take a look at our next practice problem. We have a quotient y to the fourth power over five all raised to the third power. Whenever we see quotients raised to exponents, we use the power of the quotient rule. If you remember the way this works, we're going to use the exponent of three distributed to everything that's inside the numerator and also everything that is inside the denominator and then we're going to simplify. So, on the top what we get is you're going to get y to the fourth power raised to the third power. Now, if you write it like this, it's very easy to get those powers confused. So, typically whenever you have things like this, you're gonna keep the parentheses in the numerator, and then on the bottom, it's a little bit more straightforward because all we just have is a single number raised to a power, so we don't need that parentheses there, but you could write it if you really need to. Alright? Now, on the top, we can simplify this further because we have a power that is raised to another power. We've seen this rule before. The basic idea is that when you have a power to a power, then you can so we can, multiply those exponents together. This is really like you have y to the fourth power times itself three times, and so you can multiply those exponents. On the bottom, we just have y five raised to the third power, and that's a number. Five raised to the third power is just five times five times five. So, if you simplify this, this is really just five times five times five. This is 25 times five, and this ends up being 125. Alright? So, on the bottom here, we just get 125. On the top, what ends up happening here is we get four times three for the exponent, so this just becomes 12. Why did the 12 over 125? Can we simplify this last expression any further? No, because we just have variables raised to exponents, can't simplify that. And a 125, there's nothing here we can cancel. There's no like terms or anything like that, so we can go ahead and just say that this is our most simplified answer. Alright. That's really all there is to it for, this one. Let me know if you have any questions, and I'll see you in the next

Simplify.(y45)3\left(\frac{$$y^4$$}{5}\$$right)^3$$

y12125\frac{$$y^{12}$$}{125}125y12

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6. Exponents, Polynomials, and Polynomial Functions

The Power of a Quotient Rule

Intermediate Algebra

6. Exponents, Polynomials, and Polynomial Functions

The Power of a Quotient Rule

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Multiple Choice

Simplify.
${(\frac{y^{4}}{5})}^{3}$

$\frac{y^{64}}{125}$

$$\frac{y^{12}$}{125}$

$$\frac{y^{12}$}{5}$

$\frac{y^4}{125}$

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Verified step by step guidance

Recognize that the expression is a power of a fraction: $\left(\frac{y^4}{5}\right)^3$. When raising a fraction to a power, apply the exponent to both the numerator and the denominator separately.

Apply the exponent to the numerator: $(y^4)^3$. Use the power of a power rule, which states that $(a^m)^n = a^{m \times n}$, so this becomes $y^{4 \times 3}$.

Apply the exponent to the denominator: \$5^3$. Calculate the power of the constant base.

Rewrite the expression as a single fraction with the new numerator and denominator: $\frac{y^{12}}{5^3}$.

Simplify the denominator if possible by calculating \$5^3$, and write the final simplified expression as $\frac{y^{12}}{125}$.

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Simplify.(y45)3\(\left\)(\(\frac{y^4}{5}\]\right\))^3

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Simplify.
${(\frac{y^{4}}{5})}^{3}$