To simplify the given expression [tex]\(\left(\frac{1}{2}\right)^{-3} \times\left(\frac{1}{4}\right)^{-3} \times\left(\frac{1}{5}\right)^{-3}\)[/tex], let's follow these steps:
### Step 1: Apply the Negative Exponent Rule
The negative exponent rule states that [tex]\( a^{-n} = \frac{1}{a^n} \)[/tex]. Therefore, we can rewrite each term in the product as follows:
[tex]\[
\left(\frac{1}{2}\right)^{-3} = 2^3
\][/tex]
[tex]\[
\left(\frac{1}{4}\right)^{-3} = 4^3
\][/tex]
[tex]\[
\left(\frac{1}{5}\right)^{-3} = 5^3
\][/tex]
### Step 2: Simplify Each Term
Now, calculate the powers:
[tex]\[
2^3 = 2 \times 2 \times 2 = 8
\][/tex]
[tex]\[
4^3 = 4 \times 4 \times 4 = 64
\][/tex]
[tex]\[
5^3 = 5 \times 5 \times 5 = 125
\][/tex]
### Step 3: Calculate the Product of the Simplified Terms
Finally, multiply the results obtained in step 2:
[tex]\[
8 \times 64 \times 125
\][/tex]
Given that:
[tex]\[
8 \times 64 = 512
\][/tex]
And,
[tex]\[
512 \times 125 = 64000
\][/tex]
Thus, the simplified form of the given expression [tex]\(\left(\frac{1}{2}\right)^{-3} \times\left(\frac{1}{4}\right)^{-3} \times\left(\frac{1}{5}\right)^{-3}\)[/tex] is:
[tex]\[
64000
\][/tex]
So the final answer is:
[tex]\[
\left(\frac{1}{2}\right)^{-3} \times\left(\frac{1}{4}\right)^{-3} \times\left(\frac{1}{5}\right)^{-3} = 64000
\][/tex]
