### Video Transcript

If 𝑎 is equal to the square root of seven and 𝑏 is equal to the square root of two, then what is the value of 𝑎 raised to the power of negative two multiplied by 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. To do this, we need to start by substituting 𝑎 is equal to the square root of seven and 𝑏 is equal to the square root of two into the expression. This gives us 𝑎 raised to the power of negative two multiplied by 𝑏 raised to the power of negative two is equal to root seven raised to the power of negative two multiplied by root two raised to the power of negative two.

At this point, there are a few different ways we can evaluate this expression. One way is to recall the negative exponent law, which tells us for any nonzero real number 𝑐 and exponent 𝑛, 𝑐 raised to the power of negative 𝑛 is equal to one over 𝑐 raised to the power of 𝑛. We can apply this result to each factor to obtain one over the square root of seven squared multiplied by one over the square root of two squared.

We can then recall that for any nonnegative real number 𝑐, the square root of 𝑐 all squared is just equal to 𝑐. We can use this to evaluate each factor. We get one over seven multiplied by one-half. We can then multiply the fractions by multiplying their numerators and denominators separately to get one over 14. It is worth noting that this is not the only way of evaluating this expression. We can also note that after substituting the values of 𝑎 and 𝑏 into the expression, we have a product of exponential expressions with the same exponent. This means that we can apply the power of a product rule, which tells us that if 𝑐 and 𝑑 are nonzero real numbers, then 𝑐 raised to the power of 𝑛 times 𝑑 raised to the power of 𝑛 is equal to 𝑐 times 𝑑 all raised to the power of 𝑛.

We can apply this result with 𝑐 equal to the square root of seven, 𝑑 equal to the square root of two, and 𝑛 equal to negative two to get root seven times root two all raised to the power of negative two. Since taking the square root is the same as raising the base to a power of one-half, we can apply the power of a product rule to see that if 𝑐 and 𝑑 are nonnegative real numbers, then the square root of 𝑐 times the square root of 𝑑 is equal to the square root of 𝑐 times 𝑑. This allows us to rewrite the expression as the square root of 14 all raised to the power of negative two.

We could then evaluate this expression in the same way we did before. However, there is another way that we can evaluate this expression. We could start by rewriting the square root of 14 as 14 raised to the power of one-half. We can then apply the power of a product rule, which we recall tells us that 𝑐 raised to the power of 𝑚 all raised to the power of 𝑛 is equal to 𝑐 raised to the power of 𝑚 times 𝑛. If we apply this result, then we obtain 14 raised to the power of one-half times negative two, which is equal to 14 raised to the power of negative one. Finally, we apply the negative exponent rule to see that the expression evaluates to give one over 14.