Simplify the expressions using the quotient to a power property. ( a − 3 ) 4 \left(\frac{a}{-3}\right)^4

Folks. So we have a very similar problem here, but there's one catch. We have a negative sign inside of our parentheses. We're still gonna simplify this expression by using the quotient to a power or the power of a quotient rule. Let's get started here. We're gonna take this four and distribute it to the both the numerator and the denominator. And so on the top, we're just gonna get a to the fourth power. You can't really simplify that any further. But on the bottom here, it's very, very important that you pay attention to this negative sign. We're distributing a four exponent to a bottom or a denominator of negative three. So it's very important that when you distribute this, you keep that negative sign inside the parenthesis. As we've seen before, whenever you have a term that has a negative sign that gets raised to a power, the sign will alternate based on whether the power is odd or even. So it's very, very important that you keep this in mind. So again, on the top, we can't really simplify this because yet we just have a variable to raise to an exponent. But on the bottom, we can simplify this further because this is a number raised to an exponent. If you write this out, we have eight to the fourth divided by negative three times itself four times. So if you don't really know how to deal with this negative sign, you can always just write this out to see how the negatives are gonna work out, or you can just use the rule that when you have a negative raised to an even power, the negative will cancel out. Why? Because if you take negative three times negative three, the negative goes away and it just becomes nine. And then if you have negative three times negative three again, the negative also goes away. And so it's basically like you just have nine times nine, the negatives just completely go away. And so if we simplify this one step further, this is basically like a to the fourth power divided by nine times nine. And so our final expression here is a to the fourth over 81. Just be really, really careful here. If you know the rule that if you have an even power for a negative term, the negative goes away. That's fine. Otherwise, you should just write it up and just see that the negative will just cancel itself out. Alright. So this is our final expression here, a to the fourth over 81, and we can't simplify this any further. Thanks for watching. I'll see you in the next one.

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Beginning & Intermediate Algebra

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The Power of a Quotient Rule

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Multiple Choice

Simplify the expressions using the quotient to a power property.
${(\frac{a}{- 3})}^{4}$

$\frac{a^{4}}{- 81}$

$\frac{a^{4}}{81}$

$a over 81$

$- \frac{4}{3} a$

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Verified step by step guidance

Identify the expression to simplify: $\left(\frac{a}{-3}\right)^4$.

Apply the quotient to a power property, which states that $\left(\frac{x}{y}\right)^n = \frac{x^n}{y^n}$, so rewrite the expression as $\frac{a^4}{(-3)^4}$.

Calculate the denominator by raising $-3$ to the 4th power: $(-3)^4 = (-3) \times (-3) \times (-3) \times (-3)$.

Since an even power of a negative number results in a positive number, simplify $(-3)^4$ to a positive value.

Write the simplified expression as $\frac{a^4}{81}$, noting that the negative sign disappears because of the even exponent.

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Struggling with Beginning & Intermediate Algebra?

Simplify the expressions using the quotient to a power property.(a−3)4\(\left\)(\(\frac{a}{-3}\]\right\))^4

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Simplify the expressions using the quotient to a power property.
${(\frac{a}{- 3})}^{4}$