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Simplify and provide reasons:

a) [tex]\(\left(\frac{1}{2}\right)^{-3} \times\left(\frac{1}{4}\right)^{-3} \times\left(\frac{1}{5}\right)^{-3}\)[/tex]


Sagot :

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]