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Simplify the expression π‘ to the three-eighths power times π£ to the negative five-fourths power over π‘ to the two-thirds power times π£ to the one-half power, all taken to the negative two-thirds power.
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To simplify this expression, weβll need to think about our exponent rules.
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We know that one over π₯ to the π power is equal to π₯ to the negative π power.
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And that means itβs possible for us to move π‘ to the two-thirds power and π£ to the one-half power out of the denominator and into the numerator of this expression.
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Even though π‘ and π£ have fractional exponents, we still can rewrite them with a negative fractional exponent and move them to their numerator.
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If we do that, we end up with π‘ to the negative two-thirds power and π£ to the negative one-half power.
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After that, inside the brackets, weβre multiplying exponential values with the same base.
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And so, we can say that π₯ to the π power times π₯ to the π power is equal to π₯ to the π plus π power.
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And that means we can simplify by adding three-eighths and negative two-thirds.
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To do that, they need a common denominator.
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Weβre 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 π‘.
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To find the power of our base π£, we need to add negative five-fourths plus negative one-half which will be negative five-fourth plus negative two-fourths, negative seven-fourths.
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And now, we should consider that weβre taking π‘ to the negative seven twenty-fourths times π£ to the negative seven-fourths to the negative two-thirds power.
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To take a power of a power, we say that π₯ to the π power to the π power is equal to π₯ to the π times π power.
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This means we need to multiply negative seven twenty-fourths by negative two-thirds.
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This gives us 14 over 72.
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Since both of these values are divisible by two, we can reduce that to seven over 36 so that the power for our π‘ base is seven over 36.
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And for the base π£, we need to multiply negative seven-fourths by negative two-thirds.
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This gives us fourteen twelfths.
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Again, both the numerator and the denominator are divisible by two.
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So, this fraction reduces to seven over six and becomes the power for base π£.
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So, we can say the simplified form of this expression is π‘ to the seven thirty-sixths power times π£ to the seven-sixths power.