Video: Simplifying Algebraic Expressions Using the Laws of Exponents

Simplify (π‘₯ΒΉΒ² Γ— π‘₯⁸)/(π‘₯ Γ— π‘₯ΒΉΒΉ), where π‘₯ β‰  0.

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Video Transcript

Simplify π‘₯ to the power of 12 multiplied by π‘₯ to the power of eight divided by π‘₯ multiplied by π‘₯ to the power of 11, where π‘₯ is not equal to zero.

In order to simplify this expression, we need to recall two of the laws of exponents. Firstly, π‘₯ to the power of π‘Ž multiplied by π‘₯ to the power of 𝑏 is equal to π‘₯ to the power of π‘Ž plus 𝑏. We add the exponents or indices. Also, π‘₯ to the power of π‘Ž divided by π‘₯ to the power of 𝑏 is equal to π‘₯ to the power of π‘Ž minus 𝑏. When dividing, we subtract the exponents or indices.

Let’s firstly consider the numerator of our expression, π‘₯ to the power of 12 multiplied by π‘₯ to the power of eight. As we are multiplying the two terms, we need to add the exponents. 12 plus eight is equal to 20. Therefore, the numerator simplifies to π‘₯ to the power of 20. The denominator of the original expression was π‘₯ multiplied by π‘₯ to the power of 11.

π‘₯ is the same as π‘₯ to the power of one, so we need to add one and 11. This is equal to 12. So, the denominator simplifies to π‘₯ to the power of 12. The expression π‘₯ to the power of 12 multiplied by π‘₯ to the power of eight divided by π‘₯ multiplied by π‘₯ to the power of 11 simplifies to π‘₯ to the power of 20 over π‘₯ to the power of 12.

The line in the fraction means divide, so we need to use the second law. We need to subtract the exponents, 20 minus 12. This is equal to eight. The expression in its simplest form is π‘₯ to the power of eight. We could also have solved the problem by canceling from the start. π‘₯ to the power of 12 divided by π‘₯ to the power of 11 simplifies to π‘₯ to the power of one or just π‘₯. We could then cancel the π‘₯ on the top with the π‘₯ on the bottom. This leaves us with π‘₯ to the power of eight, which was the correct answer.

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