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Problem 102

# Show that (a) $$\quad(-8)^{2 / 3}=\left(-8^{1 / 3}\right)^{2}$$ (b) $$\quad(1 / 64)^{4 / 3}=\left[(1 / 64)^{1 / 3}\right]^{4}$$

Expert verified
In summary, we have shown that the expressions are equal: (a) $$\quad(-8)^{2/3}=\left(-8^{1/3}\right)^{2}$$, where both expressions simplify to 4. (b) $$\quad(1/64)^{4/3}=\left[(1/64)^{1/3}\right]^{4}$$, where both expressions simplify to $$1/256$$.
See the step by step solution

## Step 1: Part (a): Simplify $$(-8)^{2/3}$$

The given expression is $$(-8)^{2/3}$$. The exponent $$2/3$$ can be interpreted as taking the cube root first and then squaring the result. So, let's simplify the expression step by step: $$\quad(-8)^{2/3} =[\,(-8)^{1/3}\,]^2$$ Now, calculate the cube root of $$-8$$: $$\quad(-8)^{1/3} =-2$$ Now, square the result: $$\quad(-2)^2 =4$$

## Step 2: Part (a): Simplify $$\left(-8^{1/3}\right)^{2}$$

The given expression is $$\left(-8^{1/3}\right)^{2}$$. First, find the cube root of -8: $$\quad-8^{1/3} =-2$$ Now, square the result: $$\quad(-2)^2 =4$$ Both expressions yielded the same result, so we can conclude that: $$\quad(-8)^{2/3} = \left(-8^{1/3}\right)^2$$

## Step 3: Part (b): Simplify $$(1/64)^{4/3}$$

The given expression is $$(1/64)^{4/3}$$. The exponent $$4/3$$ can be interpreted as taking the cube root first and then raising the result to the power of 4. So, let's simplify the expression step by step: $$\quad(1/64)^{4/3} =[\,(1/64)^{1/3}\,]^4$$ Now, calculate the cube root of $$1/64$$: $$\quad(1/64)^{1/3} =1/4$$ Now, raise the result to the power of 4: $$\quad(1/4)^4 =1/256$$

## Step 4: Part (b): Simplify $$\left[(1/64)^{1/3}\right]^4$$

The given expression is $$\left[(1/64)^{1/3}\right]^4$$. First, find the cube root of $$1/64$$: $$\quad(1/64)^{1/3} = 1/4$$ Now, raise the result to the power of 4: $$\quad(1/4)^4 = 1/256$$ Both expressions yielded the same result, so we can conclude that: $$\quad (1/64)^{4/3} = \left[(1/64)^{1/3}\right]^4$$

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