Video Transcript
If 𝑎 is equal to the square root of seven and 𝑏 is equal to the square root of 14, find the value of 𝑎 squared times 𝑏 squared minus 𝑎 raised to the power of negative two over 𝑏 raised to the power of negative two.
In this question, we are given the values of 𝑎 and 𝑏 and asked to evaluate an algebraic expression involving 𝑎 and 𝑏. We can evaluate the expression by substituting 𝑎 is equal to the square root of seven and 𝑏 is equal to the square root of 14 into the expression to get root seven squared times root 14 squared minus root seven raised to the power of negative two over root 14 raised to the power of negative two.
In the order of operations, we need to evaluate the exponents before the products, quotients, and differences. So, we will start by recalling that for any nonnegative real number 𝑐, the square root of 𝑐 all squared is equal to 𝑐. We can use this to evaluate root seven squared and root 14 squared. We get seven times 14 minus root seven raised to the power of negative two over root 14 raised to the power of negative two.
There are many different ways of evaluating the second term, and we will only go through one of these. We note that we have a quotient of exponential expressions raised to the same exponent. So, we can simplify the expression by using the quotient rule for exponents, which tells us 𝑐 raised to the power of 𝑛 over 𝑑 raised to the power of 𝑛 is equal to 𝑐 over 𝑑 all raised to the power of 𝑛. Applying this to the second term and evaluating the multiplication gives us 98 minus root seven over root 14 all raised to the power of negative two.
We can apply this result again by recalling that taking a square root is the same as raising the number to an exponent of one-half. So, root 𝑐 over root 𝑑 is equal to the square root of 𝑐 over 𝑑 provided 𝑐 and 𝑑 are positive. This gives us 98 minus the square root of seven over 14 all raised to the power of negative two. We can then cancel the shared factor of seven in the numerator and denominator of the radicand to obtain one-half. This gives us 98 minus the square root of one-half all raised to the power of negative two.
There are a few ways of evaluating the second term. One way is to use a specific version of the power of a power rule. We know that taking a square root is the same as raising the number to an exponent of one-half. So, we know that root 𝑐 all raised to the power of 𝑚 is equal to 𝑐 raised to the power of 𝑚 over two. Applying this result with 𝑐 equal to one-half and 𝑚 equal to negative two gives us 98 minus one-half raised to the power of negative two over two. We can then cancel the shared factor of two in the numerator and denominator of the exponent to see that our exponent is equal to negative one.
Finally, we can use a specific version of the negative exponent rule. We know that raising a nonzero base to a negative exponent is the same as raising the reciprocal of the base to the positive exponent. So, 𝑐 raised to the power of negative one is just equal to one over 𝑐 provided that 𝑐 is nonzero. This means that one-half raised to the power of negative one is equal to two. So we have 98 minus two, which we can calculate is equal to 96.
