Questions: Write without using zero or negative exponents and simplify a. (c/2)^(-6)= b. (5q)^(-2)= c. 2 * 5^(-2)= d. 5y^(-4)=

Write without using zero or negative exponents and simplify a. (c/2)^(-6)= b. (5q)^(-2)= c. 2 * 5^(-2)= d. 5y^(-4)=
Transcript text: Write without using zero or negative exponents and simplify a. $\left(\frac{c}{2}\right)^{-6}=$ $\square$ b. $(5 q)^{-2}=$ $\square$ c. $2 \cdot 5^{-2}=$ $\square$ d. $5 y^{-4}=$ $\square$
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Solution

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Solution Steps

To simplify expressions with negative exponents, we can use the property that \(a^{-n} = \frac{1}{a^n}\). We will apply this property to each part of the question and simplify accordingly.

a. \(\left(\frac{c}{2}\right)^{-6}\) can be rewritten as \(\left(\frac{2}{c}\right)^6\).

b. \((5q)^{-2}\) can be rewritten as \(\frac{1}{(5q)^2}\).

c. \(2 \cdot 5^{-2}\) can be rewritten as \(2 \cdot \frac{1}{5^2}\).

Step 1: Simplify \(\left(\frac{c}{2}\right)^{-6}\)

Using the property \(a^{-n} = \frac{1}{a^n}\), we can rewrite \(\left(\frac{c}{2}\right)^{-6}\) as: \[ \left(\frac{c}{2}\right)^{-6} = \left(\frac{2}{c}\right)^6 \] Given \(c = 1\), we calculate: \[ \left(\frac{2}{1}\right)^6 = 2^6 = 64 \]

Step 2: Simplify \((5q)^{-2}\)

Using the property \(a^{-n} = \frac{1}{a^n}\), we can rewrite \((5q)^{-2}\) as: \[ (5q)^{-2} = \frac{1}{(5q)^2} \] Given \(q = 1\), we calculate: \[ \frac{1}{(5 \cdot 1)^2} = \frac{1}{25} = 0.04 \]

Step 3: Simplify \(2 \cdot 5^{-2}\)

Using the property \(a^{-n} = \frac{1}{a^n}\), we can rewrite \(5^{-2}\) as: \[ 5^{-2} = \frac{1}{5^2} \] Thus, we have: \[ 2 \cdot 5^{-2} = 2 \cdot \frac{1}{5^2} = 2 \cdot \frac{1}{25} = 2 \cdot 0.04 = 0.08 \]

Final Answer

a. \(\left(\frac{c}{2}\right)^{-6} = \boxed{64}\)

b. \((5q)^{-2} = \boxed{0.04}\)

c. \(2 \cdot 5^{-2} = \boxed{0.08}\)

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