Simplify (-1/(2a^-2))^-3

The problem asks you to perform algebraic manipulations on a given expression, specifically an exponentiation of a fraction that contains a negative exponent. The expression to be simplified is the inverse of a fraction raised to the power of negative three. You need to apply the rules of exponents, which might involve inverting the fraction and changing the sign of the exponent to simplify the expression to its simplest form.

${((- \frac{1}{2 a^{- 2}}))}^{- 3}$

Answer

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Solution:

Step 1:

Apply the negative exponent rule to move $a^{- 2}$ to the numerator: ${(- \frac{1}{2 a^{- 2}})}^{- 3} = {(- \frac{a^{2}}{2})}^{- 3}$ .

Step 2:

Invert the fraction to change the sign of the exponent: ${(- \frac{a^{2}}{2})}^{- 3} = {(- \frac{2}{a^{2}})}^{3}$ .

Step 3:

Distribute the exponent across the fraction using the power of a product rule.

Step 3.1:

Raise $- 1$ to the third power: ${(- 1)}^{3} {(\frac{2}{a^{2}})}^{3}$ .

Step 3.2:

Raise both the numerator and the denominator to the third power separately: ${(- 1)}^{3} \frac{2^{3}}{{(a^{2})}^{3}}$ .

Step 4:

Compute the cube of $- 1$ : $- \frac{2^{3}}{{(a^{2})}^{3}}$ .

Step 5:

Calculate $2$ raised to the power of $3$ : $- \frac{8}{{(a^{2})}^{3}}$ .

Step 6:

Raise the denominator to the power of $3$ by multiplying the exponents.

Step 6.1:

Apply the power rule to multiply the exponents: $- \frac{8}{a^{2 \cdot 3}}$ .

Step 6.2:

Perform the multiplication of the exponents: $- \frac{8}{a^{6}}$ .

Knowledge Notes:

The problem-solving process involves simplifying an expression with negative and fractional exponents. Here are the relevant knowledge points and detailed explanations:

Negative Exponent Rule: For any nonzero number $b$ and integer $n$ , $b^{- n} = \frac{1}{b^{n}}$ . This rule is used to transform negative exponents into positive ones by taking the reciprocal of the base.
Power of a Product Rule: For any real numbers $a$ and $b$ , and integer $n$ , $(a b)^{n} = a^{n} b^{n}$ . This rule allows us to apply an exponent to each factor within a product separately.
Power of a Quotient Rule: For any nonzero number $a$ , real number $b$ , and integer $n$ , ${(\frac{a}{b})}^{n} = \frac{a^{n}}{b^{n}}$ . This rule is similar to the power of a product rule but applies to quotients.
Power Rule: For any real number $a$ and integers $m$ and $n$ , $(a^{m})^{n} = a^{m n}$ . This rule is used to simplify expressions with powers raised to additional powers by multiplying the exponents.

In the given problem, these rules are applied in sequence to simplify the expression $(- 1 / (2 a^{- 2}))^{- 3}$ . The solution involves inverting the fraction, distributing the exponent, and simplifying the expression step by step to reach the final simplified form.