Given that the square root of all of seven over three all to the power of 𝑥 equals seven over three, find the value of seven over three to the power of 𝑥 plus one.
So here we have our square root of seven over three to the power of 𝑥. And we’re asked to find seven over three to the power of 𝑥 plus one. Let’s start by working out what the value of 𝑥 is.
If we look at our first equation, the square root of seven over three to the power of 𝑥 equals seven-thirds. We can notice that we have both a square root and our power of 𝑥, which effectively cancel to give us seven over three. In this case, we might already have an idea of what 𝑥 could be. But let’s see if we can work it through algebraically to find 𝑥.
Let’s start by taking the value inside our parentheses and seeing if we can rewrite this in another way. We can recall that the square root of a number, for example, the square root of 𝑥, is equivalent to writing 𝑥 to the power of one-half. This means that our value of the square root of seven over three can be written as seven over three to the power of one-half.
It’s important to notice that our square root and, therefore, our to the power of half includes both the numerator and the denominator. Looking at the right-hand side, we can see that we now have a value to the power of a half and to the power of 𝑥. To help us, we can use one of our exponent rules that 𝑥 to the power of 𝑎 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 𝑏.
We can now multiply our exponents to give us seven-thirds to the power of a half times 𝑥. We can write a half times 𝑥 in a simpler way as 𝑥 over two.
Now since we were told that the square root of seven-thirds to the power of 𝑥 is equal to seven-thirds, this means that seven-thirds to the power of 𝑥 over two is equal to seven-thirds. If we look at the powers of both of our values of seven over three, we notice that seven-thirds on the right-hand side isn’t written with a power. Which means that we could write in one as our power, since seven-thirds to the power of one is equivalent to seven-thirds.
Now if we look at our equation, we notice that the seven-thirds to the power of 𝑥 over two must be equal to the seven-thirds to the power of one. This means that we can equate the exponents. That is, 𝑥 over two equals one. So 𝑥 must be equal to two.
To answer the final part of our question then, we need to find the value of seven-thirds to the power of 𝑥 plus one. We’ve just worked out that 𝑥 equals two. So our exponent will be two plus one, which is three. So let’s find the value of seven-thirds to the third power. We remember that when we take the third power of all of a fraction, that means we take the third power of the numerator and the third power of the denominator. We’ll then have seven times seven times seven over three times three times three.
Working out our numerator, our first seven times seven gives us 49. And multiplying 49 by the final seven gives us 343. On our denominator, we have three times three, which is nine, multiplied by three, giving us 27. This means that we can give our final answer as a fraction 343 over 27.