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Video: Simplifying Rational Algebraic Expressions Using Laws of Exponents

Rhodri Jones

Simplify (𝑥⁹ × 𝑥⁵ × 𝑥⁶)/(𝑥⁶ × 𝑥⁸ × 𝑥²).

02:16

Video Transcript

Simplify 𝑥 to the power of nine multiplied by 𝑥 to the power of five multiplied by 𝑥 to the power of six divided by 𝑥 to the power of six multiplied by 𝑥 to the power of eight multiplied by 𝑥 squared.

In order to simplify, the expression we need to use our laws of indices. 𝑥 to the power of 𝑎 multiplied by 𝑥 to the power of 𝑏 is equal to 𝑥 to the power 𝑎 plus 𝑏.

When we are multiplying, we can add our exponents or powers. Also, 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. When we are dividing we can subtract are exponents or powers.

If we firstly considered the numerator, 𝑥 to the power of nine multiplied by 𝑥 to the power of five multiplied by 𝑥 to the power of six, this is equal to 𝑥 to the power of 20, as nine plus five plus six is equal to 20. Adding the three exponents gives us an answer of 20.

Using the same law on the bottom 𝑥 to the power of six multiplied by 𝑥 to the power of eight multiplied by 𝑥 squared is equal to 𝑥 to the power of 16, as six plus eight plus two is equal to 16.

Our final step is to divide 𝑥 to the power of 20 by 𝑥 to the power of 16. Well our second law of indices tells us that when we divide, we can subtract the powers. 20 minus 16 is equal to four. Therefore, our answer is 𝑥 to the power of four.

By using our laws of indices the simplified version of 𝑥 to the power of nine multiplied by 𝑥 to the power of five multiplied by 𝑥 to the power of six divided by 𝑥 to the power of six multiplied by 𝑥 to the power of eight multiplied by 𝑥 squared is equal to 𝑥 to the power of four.