Video Transcript
Which of the following is equal to
the square root of seven to the power of eight divided by the square root of seven
to the power of six? The choices provided are (A) seven
to the power of four-thirds. (B) Seven squared. (C) The square root of seven to the
power of 14. (D) The square root of seven
squared. And (E) the square root of seven to
the power of four-thirds.
To find which of these expressions
is equal to our quotient, we need to simplify the quotient. As we have two powers with the same
base, where one is divided by the other, then we can use the quotient rule for
exponents, which states that 𝑎 to the power of 𝑚 divided by 𝑎 to the power of 𝑛
is equal to 𝑎 to the power of 𝑚 minus 𝑛 for a nonzero real number 𝑎 and integers
𝑚 and 𝑛.
Applying the rule with 𝑎 equal to
the square root of seven, 𝑚 equal to eight, and 𝑛 equal to six, we get the square
root of seven to the power of eight divided by the square root of seven to the power
of six is equal to the square root of seven to the power of eight minus six, which
equals the square root of seven squared. Thus, the quotient of the square
root of seven to the power of eight and the square root of seven to the power of six
is equal to the square root of seven squared, which is answer (D).
Without using the quotient rule, we
could approach this problem using repeated multiplication. In this case, we have eight factors
of the square root of seven in the numerator and six factors in the denominator. Canceling factors leaves us with
two factors of the square root of seven in the numerator, which equals the square
root of seven squared. It is also worth noting that for 𝑎
greater than zero, the square root of 𝑎 squared is equal to 𝑎. Therefore, the square root of seven
squared equals seven. But since seven is not one of the
options provided, we stay with answer (D).
