Video: Simplifying Rational Algebraic Expressions Using Laws of Exponents with Negative Exponents

Given that 𝑥 ≠ 0, simplify (𝑥⁻⁷ × 𝑥⁹)/(𝑥⁻³ × 𝑥⁻⁶).

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Video Transcript

Given that 𝑥 is not equal to zero, simplify 𝑥 to the power of negative seven multiplied by 𝑥 to the power of nine divided by 𝑥 to the power of negative three multiplied by 𝑥 to the power of negative six.

In order to solve this problem, we need to use two of the laws of exponents. Firstly, 𝑥 to the power of 𝑎 multiplied by 𝑥 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 plus 𝑏; we can add the exponents or powers. Secondly, 𝑥 to the power of 𝑎 divided by 𝑥 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 minus 𝑏, subtracting the exponents or powers.

Firstly, we can add the exponents on the top, the numerator, negative seven plus nine, so we have 𝑥 to the power of negative seven plus nine. In the same way, we can add the exponents on the bottom, the denominator, negative three plus negative six; we have 𝑥 to the power of negative three plus negative six.

Simplifying the numerator gives us 𝑥 to the power of two, as negative seven plus nine is equal to two, and on the bottom, the denominator, negative three plus negative six or negative three minus six is negative nine, so we have 𝑥 to the power of negative nine.

We can then use the second law of exponents that we wrote down, giving us 𝑥 to the power of two minus negative nine. Two minus negative nine is the same as two plus nine. Therefore, our final answer is 𝑥 to the power of 11.

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