Video: Simplifying Rational Algebraic Expressions Using Laws of Exponents

Simplify ((6^𝑦) Γ— (6^𝑦) Γ— (6^𝑦) Γ— (6^𝑦) Γ— (6^𝑦) Γ— (6^𝑦))/((6^𝑦) + (6^𝑦) + (6^𝑦) + (6^𝑦) + (6^𝑦) +(6^𝑦)).

02:54

Video Transcript

Simplify six 𝑦 times six 𝑦 times six 𝑦 times six 𝑦 times six 𝑦 times six 𝑦 over six 𝑦 plus six 𝑦 plus six 𝑦 plus six 𝑦 plus six 𝑦 plus six 𝑦.

In other words, simplify six 𝑦 times itself six times over six 𝑦 plus itself six times. This is not as simple as it might look. We need to consider our exponent rules. If we multiply π‘₯ to the π‘Ž power times π‘₯ to the π‘Ž power, we get π‘₯ to the π‘Ž power squared. We take a power of a power. We can apply this to our numerator. Instead of having six 𝑦 times itself six times, we can write six 𝑦 to the sixth power. This will be the new numerator, six 𝑦 to the sixth power.

The denominator is a little different. Consider π‘₯ to the π‘Ž power plus π‘₯ to the π‘Ž power. We say that’s two π‘₯ to the π‘Ž power. We have two π‘₯ to the π‘Ž power terms. In our case, we have six six to the 𝑦 power terms. And we can break this up. We can say that divided by six is the same thing as multiplying by one-sixth. And then, we’ll have six to the 𝑦 power to the sixth power over six to the 𝑦 power.

In the denominator, we could call six to the 𝑦 power six to the 𝑦 power to the first power. And that would cancel out one of the six to the 𝑦 power terms. Our numerator then becomes six to the 𝑦 power to the fifth power.

Okay, what can we say about this one over six? We can call it one over six to the first power. And we’re going to multiply that by six to the 𝑦 power to the fifth power. We can rewrite one over six to the first power as six to the negative one power. And we can take this power to a power six to the 𝑦 to the fifth power and rewrite it as six to the five 𝑦 power.

When final rule will help us with our simplification, when our exponents have the same base and they’re being multiplied together, we add their exponents. We will have six to the negative one plus five 𝑦 power, which we could rewrite as six to the five 𝑦 minus one power.

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