Which of the following has the same value as two-thirds squared divided by two- thirds cubed? Is it A) three over two, B) two over three, C) 32 over 243, D) four over nine to the power of five, or E) one to the power of two-thirds.
As the fractions in the brackets or parentheses are the same, we can use one of the laws of indices or exponents: 𝑥 to the power of 𝑎 divided by 𝑥 to the power of 𝑏 is equal to 𝑥 to the power of 𝑎 minus 𝑏. We can subtract the exponents or indices. We can simplify two-thirds squared divided by two-thirds cubed by subtracting the indices two and three. Two minus three is equal to negative one. Therefore, two-thirds squared divided by two-thirds cubed is equal to two-thirds to the power of negative one.
Any value to the power of negative one is equal to one over that value. The power negative one means the reciprocal. The reciprocal of any fraction is the fraction written upside down. This means that two-thirds to the power of negative one is equal to three over two or three-halves. The numerator has become the denominator and vice versa. This is because one divided by two-thirds is equal to three-halves. This means that the correct answer is option A. Three over two or three-halves has the same value as two- thirds squared divided by two-thirds cubed.