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Math  /  Algebra

Question3. (8−34)−89\left(8^{\frac{-3}{4}}\right)^{\frac{-8}{9}}

Studdy Solution

STEP 1

What is this asking? Simplify the expression (8−34)−89\left(8^{\frac{-3}{4}}\right)^{\frac{-8}{9}} to its simplest form. Watch out! Don't forget the rules of exponents, especially when dealing with negative and fractional exponents!

STEP 2

1. Simplify the inner exponent
2. Apply the outer exponent
3. Simplify the final expression

STEP 3

Alright, let's start by looking at the expression inside the parentheses: 8−348^{\frac{-3}{4}}.
The exponent −34\frac{-3}{4} means we're dealing with a **negative exponent**, which tells us to take the reciprocal of the base.
So, we have:
8−34=18348^{\frac{-3}{4}} = \frac{1}{8^{\frac{3}{4}}}

STEP 4

Now, let's deal with the fractional exponent 34\frac{3}{4}.
This can be split into a root and a power.
The denominator, 4, indicates a fourth root, and the numerator, 3, indicates a cube.
So we have:
834=(84)38^{\frac{3}{4}} = \left(\sqrt[4]{8}\right)^3

STEP 5

Since 8=238 = 2^3, we can rewrite 84\sqrt[4]{8} as 234\sqrt[4]{2^3}.
Using the property of exponents, this becomes:
234=234\sqrt[4]{2^3} = 2^{\frac{3}{4}}
So,
834=(234)3=2948^{\frac{3}{4}} = \left(2^{\frac{3}{4}}\right)^3 = 2^{\frac{9}{4}}

STEP 6

Putting it all together, we have:
8−34=1294=2−948^{\frac{-3}{4}} = \frac{1}{2^{\frac{9}{4}}} = 2^{-\frac{9}{4}}

STEP 7

Now, let's apply the outer exponent −89\frac{-8}{9} to the expression 2−942^{-\frac{9}{4}}:
(2−94)−89\left(2^{-\frac{9}{4}}\right)^{\frac{-8}{9}}

STEP 8

When raising a power to another power, we multiply the exponents:
2−94⋅−89=272362^{-\frac{9}{4} \cdot \frac{-8}{9}} = 2^{\frac{72}{36}}

STEP 9

Simplify the exponent 7236\frac{72}{36}:
7236=2\frac{72}{36} = 2

STEP 10

So, the expression simplifies to:
22=42^2 = 4

STEP 11

The simplified form of the expression (8−34)−89\left(8^{\frac{-3}{4}}\right)^{\frac{-8}{9}} is 4\boxed{4}.

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