Transcription
If the square root of x times the square root of x equals x to the power of a, what's a? You have 5 seconds.
When most people see roots inside of roots, they panic. They either guess or they try to square both sides over and over again until they lose track of all the exponents and they create some crazy algebraic mess. But, you don't have to square anything.
A square root is just a disguise for a fractional exponent. Specifically, it's a power of 1/2. Let's break it down from the inside out. Look at that inner square root of x. We can just rewrite that as x to the power of 1/2. Now, multiply by the normal x right next to it. Remember your rules. When you multiply the same base, you add the exponents. 1 + 1/2 is 3/2. So, the entire inside just goes to x to the power of 3/2.
Now, for the big outer roof, that massive square root wrapping the whole thing, that is just another 1/2 power wrapping our new term. When you raise a power to a power, you multiply them. 3/2 * 1/2 is 3/4. So, a equals 3/4.
And if you don't believe the algebra, let's look at the graph. Here is the very strange curve for our nested square root equation. And here is the clean curve for x to the power of 0.75. They're the exact same line. The SAT tries to scare you with notation. Don't let them.