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

Kathryn Kingham

Write 7⁵ᐟ¹² × 7³ᐟ¹² in radical form.


Video Transcript

Write seven to the five twelfths power times seven to the three twelfths power in radical form.

We wanna convert these rational exponents into radical form. To convert rational exponents into radical form, we use this formula: 𝑥 to 𝑚 over 𝑛 equals the 𝑛 route of 𝑥 to the 𝑚 power. We’re multiplying two exponents with the same base. This means that our first step here is to add the two exponents together to simplify. This gives us seven to the eight twelfths power.

But I can actually reduce this exponent because it’s not a fraction in simplest form. If we divide our numerator and our denominator by four here, we’re left with seven to the two-thirds power.

Okay. Now we need this guy. In our example, seven to the two-thirds, our 𝑥 is equal to seven, our 𝑚 is equal to two, and our 𝑛 is equal to three. The 𝑥-value in our case, seven, goes inside the radical. The 𝑛-value represents our root, which in our case is three. And the 𝑚-value for us, two, is the exponent that’s outside the parentheses.

Radical form of this expression looks like this: the cube root of seven squared.