Video Transcript
Simplify the expression 𝑡 to the three-eighths power times 𝑣 to the negative five-fourths power over 𝑡 to the two-thirds power times 𝑣 to the one-half power, all taken to the negative two-thirds power.
To simplify this expression, we’ll need to think about our exponent rules. We know that one over 𝑥 to the 𝑎 power is equal to 𝑥 to the negative 𝑎 power. And that means it’s possible for us to move 𝑡 to the two-thirds power and 𝑣 to the one-half power out of the denominator and into the numerator of this expression. Even though 𝑡 and 𝑣 have fractional exponents, we still can rewrite them with a negative fractional exponent and move them to their numerator. If we do that, we end up with 𝑡 to the negative two-thirds power and 𝑣 to the negative one-half power.
After that, inside the brackets, we’re multiplying exponential values with the same base. And so, we can say that 𝑥 to the 𝑎 power times 𝑥 to the 𝑏 power is equal to 𝑥 to the 𝑎 plus 𝑏 power. And that means we can simplify by adding three-eighths and negative two-thirds. To do that, they need a common denominator. We’re combining nine twenty-fourths and negative sixteen twenty-fourths, which will give us negative seven twenty-fourths as the new power value for the exponent with the base 𝑡.
To find the power of our base 𝑣, we need to add negative five-fourths plus negative one-half which will be negative five-fourth plus negative two-fourths, negative seven-fourths. And now, we should consider that we’re taking 𝑡 to the negative seven twenty-fourths times 𝑣 to the negative seven-fourths to the negative two-thirds power.
To take a power of a power, we say that 𝑥 to the 𝑎 power to the 𝑏 power is equal to 𝑥 to the 𝑎 times 𝑏 power. This means we need to multiply negative seven twenty-fourths by negative two-thirds. This gives us 14 over 72. Since both of these values are divisible by two, we can reduce that to seven over 36 so that the power for our 𝑡 base is seven over 36. And for the base 𝑣, we need to multiply negative seven-fourths by negative two-thirds. This gives us fourteen twelfths. Again, both the numerator and the denominator are divisible by two. So, this fraction reduces to seven over six and becomes the power for base 𝑣.
So, we can say the simplified form of this expression is 𝑡 to the seven thirty-sixths power times 𝑣 to the seven-sixths power.
