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Question Video: Simplifying an Expression Involving Division of Positive Integer Powers over the Real Numbers Mathematics • 9th Grade

What is the value of (−√3)⁶ ÷ (√3)³?

02:09

Video Transcript

What is the value of negative root three to the sixth power divided by root three to the third power?

The bases in this expression are nearly the same, but one is the negative of the other. We can’t yet apply laws of exponents to simplify this division as the bases aren’t exactly the same. So let’s instead think about how we can alter this expression so that the bases are identical.

In the first term, we have the power of a product because negative root three is equal to negative one multiplied by root three. We can therefore recall the power of a product rule for exponents, which states that a product 𝑎𝑏 to the 𝑛th power is equal to 𝑎 to the 𝑛th power multiplied by 𝑏 to the 𝑛th power. So we can rewrite the first term as negative one to the sixth power multiplied by root three to the sixth power. We can then recall that negative one to any even power is positive one. So in fact, the first term simplifies to root three to the sixth power.

We now have exactly the same base for both parts of the quotient. So we can simplify using the quotient rule for exponents, which tells us that to divide powers of the same base, we subtract the exponents. So root three to the sixth power divided by root three to the third power is root three to the power of six minus three, which is root three cubed. We can express this longhand as root three multiplied by root three multiplied by root three. And then we can group the first two terms in the product together to give root three squared multiplied by root three.

We then recall that for nonnegative real values of 𝑎, the square root of 𝑎 squared is equal to 𝑎. And so the square root of three squared is equal to three. And the expression simplifies to three root three. We found that the value of negative root three to the sixth power divided by root three to the third power is three root three.

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