Math  /  Algebra

Question2. Write 322532^{-\frac{2}{5}} without using exponents or radicals.

Studdy Solution

STEP 1

What is this asking? We need to simplify 322532^{-\frac{2}{5}} and express it as a regular number without any exponents or radicals. Watch out! Negative and fractional exponents can be tricky!
Don't mix up the order of operations.

STEP 2

1. Rewrite the base
2. Apply the negative exponent
3. Apply the fractional exponent

STEP 3

We can **rewrite** 32 as 252^5.
This will help us deal with the fractional exponent later.
So, our expression becomes (25)25(2^5)^{-\frac{2}{5}}.
This is awesome because now we have a power of a power!

STEP 4

A **negative exponent** means we take the reciprocal.
So, (25)25(2^5)^{-\frac{2}{5}} becomes 1(25)25\frac{1}{(2^5)^{\frac{2}{5}}}.
This makes the exponent positive, which is much easier to work with.

STEP 5

Remember the **power of a power** rule? (am)n=amn(a^m)^n = a^{m \cdot n}.
Let's apply it here: 1(25)25\frac{1}{(2^5)^{\frac{2}{5}}} becomes 12525\frac{1}{2^{5 \cdot \frac{2}{5}}}.

STEP 6

We can **simplify** the exponent in the denominator: 5255 \cdot \frac{2}{5}.
Notice how the 5 in the numerator and the 5 in the denominator divide to one, leaving us with just 2.
So, our expression becomes 122\frac{1}{2^2}.

STEP 7

Finally, 222^2 is just 22=42 \cdot 2 = 4.
So, our **final answer** is 14\frac{1}{4}.

STEP 8

322532^{-\frac{2}{5}} simplifies to 14\frac{1}{4}.

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