Video: Multiplying Three Numbers with Rational Exponents and Expressing the Result in Radical Form

Which the following expressions is equivalent to 13^(1/9) × 13^(2/9) × 13^(3/9)? [A] (the cube root of 13)² [B] (the sixth root of 39)⁹ [C] (the square root of 13)³ [D] (the ninth root of 39)⁶ [E] (the sixth root of 13)⁹.

03:13

Video Transcript

Which the following expressions is equivalent to 13 to the power of one over nine multiplied by 13 to the power of two over nine multiplied by 13 to the power of three over nine? The options are A] the cube root of 13 all squared B], the sixth root of 39 to the power of nine, C] the square root of 13 cubed, D] the ninth root of 39 to the power of six, or E] the sixth root of 13 to the power of nine.

So to enable us to solve this problem, what we’re gonna use is some exponent rules. And the first of these rules is shown here. So we’ve got the same base, which is 𝑥. So we got to 𝑥 to the power of 𝑎 multiplied by 𝑥 to the power of 𝑏. This is equal 𝑥 to the power of 𝑎 plus 𝑏. We can see that in our expression we have the same base, because the base is 13 in each term. So therefore, if we apply the rule to our expression, we’re gonna have 13 to the power of one over nine plus two over nine plus three over nine cause we’ve added the exponents. And this will give us 13 to the power of six over nine. And that’s because we have the same denominator in each of the fractions, so we just add the numerators. But now what we can do is we can simplify the exponent because we can divide the numerator and denominator of our exponent by three because it’s a factor of both six and nine. And when we do that, we get 13 to power of two-thirds or two over three.

But if I look at the answers that we’ve been given — A, B, C, D, and E — it doesn’t match any of them. So what do I do now? Well what we do is we apply another exponent rule. And this is if we have 𝑥 to the power of 𝑎 and this is all to the power 𝑏, then it’s the same as 𝑥 to the power of 𝑎𝑏. So we’ve multiplied the exponents. Well if we look to our term. We’ve got 13 to the power of two-thirds. Well two-thirds can be made up with a third multiplied by two. So therefore, using the exponent rule, we can rewrite this as 13 to power of third then all squared. Well again when take a look back at A, B, C, D, and E. And does it look like any of the answers now?

Well, no, we’re not quite there. We can see that it’s starting to look like A, because we can see we’ve got 13 inside our parentheses, and then we’ve got two as the exponent outside the parentheses. But it’s still not quite the same. So therefore, what we need to do now is to use one more exponent rule. And that’s if we have 𝑥 the power of a third, this is equal to the cube root of 𝑥. So therefore, if we apply this exponent rule, we can rewrite 13 to the power of a third as the cube root of 13. So then we’ve got the cube root of 13 all squared. So we can say that the answer equivalent to 13 to power of one over nine multiplied by 13 to the power of two over nine multiplied by 13 to power of three over nine is A, which is the cube root of 13 all squared.

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