Video: Simplifying Rational Algebraic Expressions Using Laws of Exponents

Simplify ((√2)^(8𝑛 + 10) Γ— 30^(8𝑛 + 3))/((√2)^(8𝑛) Γ— 6^(8𝑛) Γ— 5^(8𝑛 + 4)).

03:52

Video Transcript

Simplify the square root of two to the eight 𝑛 plus 10 power times 30 to the eight 𝑛 plus three power all over the square root of two to the eight 𝑛 power times six to the eight 𝑛 power times five to the eight 𝑛 plus four power.

As we copy down this problem, I’m going to write the square root of two down as two to the one-half power since those values are equal. So, we’ll have two to the one-half power to the eight 𝑛 plus 10 power. So, the only change we made when copying everything down was to rewrite the square root of two as two to the one-half power.

To simplify this expression, one rule will be really important. And that’s π‘₯ to the π‘Ž plus 𝑏 power is equal to π‘₯ to the π‘Ž power times π‘₯ to the 𝑏 power. This means that two to the one-half power to the π‘Žπ‘› plus 10 power can be written as two to the one-half power to the eight 𝑛 times two to the one-half power to the 10th power. We’ll do the same thing for 30 to the eight 𝑛 plus three power. We’re breaking these up so that we can work on simplifying.

The 30 to the eight 𝑛 plus three power becomes 30 to the eight 𝑛 power times 30 cubed. We’ll just bring across two to the one-half power to the eight 𝑛 power in the denominator and six to the eight 𝑛. And we’ll break up five to the eight 𝑛 plus four power into five to the eight 𝑛 power times five to the fourth power. At this point, we have two to the one-half power to the eight 𝑛 power in the numerator and the denominator. And so, they cancel out.

It might seem like there’s nothing else we can cancel out. However, we have six to the eight 𝑛 power and five to the eight 𝑛 power. Six times five is 30, and so we can rewrite six to the eight 𝑛 power times five to the eight 𝑛 power as 30 to the eight 𝑛 power. Then, bring over the five to the fourth power. And from there, we’ll need to note that π‘₯ to the π‘Ž power to the 𝑏 power can be rewritten as π‘₯ to the π‘Ž times 𝑏 power. And that means two to the one-half power to the 10th power can be rewritten as two to the one-half times 10 power, which is two to the fifth power.

Now, we have two to the fifth power times 30 to the eight 𝑛 power times 30 cubed all over 30 to the eight 𝑛 power times five to the fourth power. 30 to the eight 𝑛 power in the numerator and the denominator cancels out. And now we have two to the fifth power times 30 cubed over five to the fourth power. There’s one final thing we can simplify. If we remember that 30 is five times six, we can write 30 cubed as five times six cubed, which is equal to five cubed times six cubed.

If we bring everything else over, we see that we have five cubed in the numerator and five to the fourth power in the denominator. This means that the five cubed in the numerator can be cancelled out, and five to the fourth power in the denominator becomes five to the first power. So, we have six cubed times two to the fifth power over five. Six cubed equals 216, two to the fifth power is 32, all over five. And there’s nothing else that we can simplify. So, we just multiply 216 by 32, which gives us 6912 over five.

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.