Simplify the expression. 𝑡 to the power of negative three-eighths multiplied by 𝑣 to the power of five-quarters divided by 𝑡 to the power of negative two-thirds multiplied by 𝑣 to the power of negative one-half.
In order to answer this question, we need to recall one of our laws of exponents or indices. 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. When dividing terms with the same base, we can subtract the exponents. We need to treat our two variables 𝑡 and 𝑣 separately.
Let’s begin by dividing 𝑡 to the power of negative three-eighths by 𝑡 to the power of negative two-thirds. This is the same as 𝑡 to the power of negative three-eighths minus negative two-thirds. We need to add two-thirds to negative three-eighths. When adding and subtracting fractions, we need to ensure we have a common denominator. The lowest common multiple of eight and three is 24. So, our common denominator is 24.
We have multiplied the denominator of the first fraction by three, so we also need to multiply the numerator by three. This gives us negative nine. We multiplied the denominator of the second fraction by eight. Two multiplied by eight is 16. As the denominators are now the same, we can add the numerators. Negative nine plus 16 is equal to seven. Therefore, negative three-eighths plus two-thirds is equal to seven over 24 or seven twenty-fourths. The 𝑡-term in our expression simplifies to 𝑡 to the power of seven over 24.
We now need to repeat this process with the 𝑣-terms. 𝑣 to the power of five over four divided by 𝑣 to the power of negative a half is equal to 𝑣 to the power of five over four minus negative a half. In this case, we need to add five over four or five-quarters to one-half. One-half is the same as two-quarters. This means that five-quarters plus one-half is seven-quarters. The 𝑣 part of our expression simplifies to 𝑣 to the power of seven over four.
The simplified version of our expression is 𝑡 to the power of seven over 24 multiplied by 𝑣 to the power of seven over four.