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Math  /  Algebra

Question(23)−5(2 \sqrt{3})^{-5}

Studdy Solution

STEP 1

1. We are dealing with an expression that involves an exponent and a radical.
2. The expression can be simplified by applying the properties of exponents and radicals.

STEP 2

1. Apply the negative exponent rule.
2. Simplify the expression using properties of exponents.
3. Calculate the final value.

STEP 3

Apply the negative exponent rule, which states that a−n=1an a^{-n} = \frac{1}{a^n} .
(23)−5=1(23)5 (2 \sqrt{3})^{-5} = \frac{1}{(2 \sqrt{3})^5}

STEP 4

Simplify the expression (23)5(2 \sqrt{3})^5 by applying the property (ab)n=an×bn(ab)^n = a^n \times b^n.
(23)5=25×(3)5 (2 \sqrt{3})^5 = 2^5 \times (\sqrt{3})^5

STEP 5

Calculate 252^5.
25=32 2^5 = 32

STEP 6

Simplify (3)5(\sqrt{3})^5 using the property (a)n=an/2(\sqrt{a})^n = a^{n/2}.
(3)5=35/2 (\sqrt{3})^5 = 3^{5/2}

STEP 7

Combine the results from STEP_3 and STEP_4.
(23)5=32×35/2 (2 \sqrt{3})^5 = 32 \times 3^{5/2}

STEP 8

Substitute back into the expression from STEP_1.
1(23)5=132×35/2 \frac{1}{(2 \sqrt{3})^5} = \frac{1}{32 \times 3^{5/2}}
This is the simplified form of the original expression.

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