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Problem 107

# If $$a=3$$ and $$b=2$$, find $$(6 a-b)^{-5 / 4}$$

Expert verified
$$\frac{1}{32}$$
See the step by step solution

## Step 1: Substitute the given values

We are given that $$a=3$$ and $$b=2$$. We start by substituting these values into the given expression $$(6a-b)^{-5/4}$$.

## Step 2: Simplify the expression

After substituting values for $$a$$ and $$b$$, we get: $(6(3)-2)^{-5/4}$ Now, simplifying the expression inside the parentheses: $(18-2)^{-5/4}$ $(16)^{-5/4}$

## Step 3: Evaluate the power

Now we need to evaluate the power with a negative exponent and a fraction. We can do this by breaking down the power into two parts. The $$-5/4$$ power can be thought of as first raising the expression to the reciprocal of 4 and then raising to the power of -5. $\left(16^{\frac{1}{4}}\right)^{-5}$ Now, $$16^{\frac{1}{4}}$$ means the fourth root of 16, which is 2: $(2)^{-5}$ Finally, we can evaluate the negative exponent. The power of -5 means finding the reciprocal: $\frac{1}{2^5}$ And simplify the expression: $\frac{1}{32}$ So the final answer is: $(6a-b)^{-5 / 4} = \frac{1}{32}$

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