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Answer :
To solve the expression [tex]D = \frac{243 \times 625 \times 128}{64 \times 27 \times 125}[/tex], we'll break it down step-by-step.
First, we'll look at simplifying both the numerator and the denominator separately:
Numerator: [tex]243 \times 625 \times 128[/tex]
- [tex]243[/tex] can be expressed as [tex]3^5[/tex].
- [tex]625[/tex] can be expressed as [tex]5^4[/tex].
- [tex]128[/tex] can be expressed as [tex]2^7[/tex].
So, the numerator can be rewritten using prime factorization:
[tex]243 \times 625 \times 128 = 3^5 \times 5^4 \times 2^7[/tex]Denominator: [tex]64 \times 27 \times 125[/tex]
- [tex]64[/tex] can be expressed as [tex]2^6[/tex].
- [tex]27[/tex] can be expressed as [tex]3^3[/tex].
- [tex]125[/tex] can be expressed as [tex]5^3[/tex].
So, the denominator can be rewritten as:
[tex]64 \times 27 \times 125 = 2^6 \times 3^3 \times 5^3[/tex]
Now, substitute these into the original expression:
[tex]D = \frac{3^5 \times 5^4 \times 2^7}{2^6 \times 3^3 \times 5^3}[/tex]
Next, we'll simplify by canceling out the common factors in the numerator and the denominator:
- For [tex]3^5 / 3^3[/tex], subtract the exponents: [tex]5 - 3 = 2[/tex], so we get [tex]3^2[/tex].
- For [tex]5^4 / 5^3[/tex], subtract the exponents: [tex]4 - 3 = 1[/tex], so we get [tex]5^1[/tex].
- For [tex]2^7 / 2^6[/tex], subtract the exponents: [tex]7 - 6 = 1[/tex], so we get [tex]2^1[/tex].
Thus, we have:
[tex]D = 3^2 \times 5^1 \times 2^1[/tex]
Now calculate the simplified exponents:
- [tex]3^2 = 9[/tex]
- [tex]5^1 = 5[/tex]
- [tex]2^1 = 2[/tex]
Finally, multiply them together:
[tex]D = 9 \times 5 \times 2 = 90[/tex]
Therefore, the simplified value of [tex]D[/tex] is [tex]90[/tex].
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