Question

sqrt[4 ]{8} \times \sqrt[4]{10} " alt=" \sqrt[4 ]{8} \times \sqrt[4]{10} " align="absmiddle" class="latex-formula">

Answer 1

Disclaimer: I prefer Method A because I'm used to it but I feel like Method B is easier for most people.

Method A: taking apart all the pieces and putting it back together neatly:

First step: rewrite each of the square roots as something raised to a fraction so that we don't have any square root symbols because they're kinda gross :D.

The first square root can be rewritten as 8^(1/4) and the second can be rewritten as 10^(1/4). At first it looks like nothing happened, because if you wanna get anything done with exponents and multiplication, you need a common base. Let's split 8 and 10 apart then.

8^(1/4)=(2^3)^(1/4)
Using the power of powers rule, multiply 3 and (1/4):
=2^(3/4)

10^(1/4)=2^(1/4) * 5^(1/4)

Multiply the two together: 2^(3/4)*2^(1/4)*5^(1/4)
Use the product of powers rule since 2^(3/4) and 2^(1/4) have a common base of 2.
2^(3/4+1/4)*5^(1/4)=2 ^1 *5^(1/4)=2 (tiny 4 above the left side of a square root symbol with a 5 under the square root)

Method B: building everything up into a mess and then straightening it out

The square roots have the same power (they both have a tiny 4 on the left side). So bring the 8 and 10 together under one square root, keeping the tiny 4 there. 8 times 10 is 80 so you have a square root with a tiny 4 and 80 underneath the square root symbol. That 4 raises whatever is under the square root to the (1/4) power, which means we have to find a factor of 80 that is the fourth power of a number (a.k.a. The perfect square of a perfect square). Just start listing out factors of 80: 2, 4, 5, 8, 10, 16, 20.... Wait! 16 is also 2^4. That's what we're looking for. When we can rewrite the 80 under the square root as 16*5 and take 16^(1/4) out, so writing 2 on the outside. This leaves 5 under the square root with the tiny 4 (still don't know the proper terminology for that; I'm hoping you know what I mean).

The answer is the same: 2 next to (5 to the fourth root)

Hope that helped! :)