Video Transcript
Which of the following expressions has the same value as 𝑎 over 𝑏 raised to the zeroth power times 𝑎 over 𝑏 raised to the power of negative two divided by 𝑎 over 𝑏 raised to the power of negative five all raised to the power of negative seven? Is it option (A) 𝑎 over 𝑏 raised to the power of negative seven all raised to the power of negative seven? Option (B) 𝑎 over 𝑏 raised to the seventh power all cubed. Option (C) 𝑎 over 𝑏 raised to the seventh power all raised to the power of negative seven. Option (D) 𝑎 over 𝑏 raised to the power of negative seven all cubed. Or is it option (E) 𝑎 over 𝑏 raised to the power of negative seven all raised to the fifth power?
In this question, we need to determine which of five given expressions has the same value as a given expression. To answer this question, we can start by noting that the expression we are given involves a base of 𝑎 over 𝑏 raised to various exponents. And all of our options include a base of 𝑎 over 𝑏. Therefore, we can answer this question by simplifying the given expression while keeping its base as 𝑎 over 𝑏.
There are many ways that we can simplify this expression. We can start by noting that we have a base raised to an exponent of zero. We can simplify this by recalling the zero exponent rule, which tells us that raising a nonzero base to the power of zero gives us one. We can use this to simplify our expression by replacing this factor with one to obtain the following expression. This will hold true provided 𝑎 is nonzero, but we can assume that both 𝑎 and 𝑏 are nonzero since we know that the expression is well defined. Of course, multiplying by one does not change the value of this expression, so we do not need to include this factor.
We can now notice that inside the parentheses, we have a quotient of the same base raised to different exponents. We can simplify this by recalling that the quotient rule for exponents tells us that 𝑥 raised to the power of 𝑚 over 𝑥 raised to the power of 𝑛 is 𝑥 raised to the power of 𝑚 minus 𝑛. We can use this to simplify the expression inside the parentheses by raising the base to the difference in the exponents. We obtain 𝑎 over 𝑏 raised to the power of negative two minus negative five all raised to the power of negative seven. We can then evaluate the expression in the exponent. We find that negative two minus negative five is equal to three. Therefore, we have 𝑎 over 𝑏 cubed all raised to the power of negative seven.
We can note that this is not yet exactly the same as any of the given options. This means we need to rewrite this expression further. We can do this by noting that we have a base raised to an exponent all raised to another exponent. This is the same form as the power rule for exponents that tells us 𝑥 raised to the power of 𝑚 all raised to the power of 𝑛 is equal to 𝑥 raised to the power of 𝑚 times 𝑛. Therefore, we can rewrite this expression by raising the base to the power of the product of the exponents. We obtain 𝑎 over 𝑏 all raised to the power of three times negative seven. We can then calculate that this is equal to 𝑎 over 𝑏 all raised to the power of negative 21.
We can now compare this expression to the five given options by considering the product of their exponents. The expression in option (A) is equal to 𝑎 over 𝑏 all raised to the power of negative seven times negative seven, which is 𝑎 over 𝑏 all raised to the power of 49. Since we want the expressions to be equal for all values of 𝑎 and 𝑏, we can recall that for exponential expressions of the same base to be equal, in general, their powers must be equal. So, because the powers are not equal, option (A) cannot be correct.
We can follow the same process for the other options. In option (B), we have 𝑎 over 𝑏 all raised to the power of 21, which is not a power of negative 21, so this option is incorrect. In option (C), we have 𝑎 over 𝑏 all raised to the power of negative 49. And in option (E), we have 𝑎 over 𝑏 all raised to the power of negative 35. So these are both incorrect.
Finally, in option (D), we have 𝑎 over 𝑏 all raised to the power of negative 21, which is equivalent to our expression. It is also worth noting we could have directly found this by recalling that 𝑥 raised to the power of 𝑚 all raised to the power of 𝑛 is equal to 𝑥 raised to the power of 𝑛 all raised to the power of 𝑚. In either case, we have shown that the given expression is only equivalent to option (D), 𝑎 over 𝑏 raised to the power of negative seven all cubed.
