Rewrite the expression using exponent rules.(3x4y−2)3\left(\frac{3x^4}{y^{-2}}\right)^3(y−23x4)3

27 x 12 y 6 27x^{12}y^6 27 x 12 y 6

Rewrite the expression using exponent rules. ( 3 x 4 y − 2 ) 3 \left(\frac{3x^4}{y^{-2}}\right)^3 ( y − 2 3 x 4 ) 3

Welcome back, everyone. In this practice problem, we have an expression 3x to the 4th power over y to the negative two power, and that whole entire fraction there is raised to the third power. So let's get started. Which of the rules we're gonna use? Well, if you have your rule sheet handy, you should know that anytime you have an exponent on the outside of a parenthesis, then we can use the power of a quotient rule, especially if it's a fraction. Right? So this is gonna be the power of a quotient. So what happens? Well well, with the power of a quotient rule, we distribute this 3 into everything that is inside of the parenthesis. Everything. So that means we're gonna do the 3 over here to the 3rd power. So we have the number 3 raised to the exponent of 3, and then we have the x to the 4th power to the 3rd power. That whole thing is in the numerator. And what does the denominator become? It becomes y to the negative two power, then to the 3rd power. So let's clean up the exponents a little bit in the parentheses, because we can keep simplifying this. What does the top become? Well, what happens is 3 to the third power just ends up being a number. 3 times 3 times 3 times 3 is 27, so that's what that works out to. Then we have x to the 4th power to the 3rd power. So in other words, this really is just the power rule. So this is a power on top of a power, so that's the power rule over here. In fact, both of these things on the numerator and denominator are both power rules. So what happens is we have 27, and then the top becomes x to the 12th power, and on the bottom what we have is we have y to the negative two power, and then that is to the third power. So when we multiply these two numbers, remember what happens is we're gonna get y to the negative 6th power. You multiply the exponents. So can this be simplified any further? Well, usually, what happens is we when we have negative exponents, we don't wanna leave them like that. So remember what happens with negative exponents is they actually get flipped to the top. If they're on the bottom, they go to the top, and we write them with a positive exponent. So this whole thing actually just becomes 27 x to the 12th power, y to the 6th power. So this goes up here, and this is because this is a negative exponent. So we use the negative exponent rule. Alright? So hopefully this makes sense. This expression can't be simplified anymore. Thanks for watching. See you in the next one.

Rewrite the expression using exponent rules. ( 3 x 4 y − 2 ) 3 \left(\frac{3x^4}{y^{-2}}\right)^3 ( y − 2 3 x 4 ) 3

27 x 12 y 6 27x^{12}y^6 27 x 12 y 6

3 x 12 y 6 3x^{12}y^6 3 x 12 y 6

27 x 12 y 6 \frac{27x^{12}}{y^6} y 6 27 x 12

27 x 12 y 2 27x^{12}y^2 27 x 12 y 2

0. Functions

Exponent rules

Calculus

0. Functions

Exponent rules

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Multiple Choice

Rewrite the expression using exponent rules.
$\left(\frac{3x^4}{y^{-2}}\right)^3$

$3x^{12}y^6$

$\frac{27x^{12}}{y^6}$

$27x^{12}y^2$

$27x^{12}y^6$

Verified step by step guidance

Start by applying the power of a quotient rule, which states that $ \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} $. Apply this to the expression $ \left( \frac{3x^4}{y^{-2}} \right)^3 $.

Raise each part of the fraction to the power of 3: $ (3x^4)^3 $ and $ (y^{-2})^3 $.

Use the power of a power rule, which states that $ (a^m)^n = a^{m \cdot n} $. Apply this to $ (3x^4)^3 $ to get $ 3^3 \cdot (x^4)^3 = 27x^{12} $.

Similarly, apply the power of a power rule to $ (y^{-2})^3 $ to get $ y^{-6} $.

Combine the results to rewrite the expression as $ \frac{27x^{12}}{y^{-6}} $. Since $ y^{-6} = \frac{1}{y^6} $, the expression simplifies to $ 27x^{12}y^6 $.

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