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Solve the following system using three different methods. You will need to upload a picture clearly showing all three methods.

[tex]\[
\frac{\left(4 a^3 b^8 c^5 d^{-1} e f^{-4}\right)^2\left(19 a b c d^4\right)^0}{a^3 b^{-9} c d^{-1}}
\][/tex]

Sagot :

GinnyAnswer
Sure, let's simplify the given expression step-by-step using algebraic rules. For each step, we'll make sure we clearly understand each part of the process.

First, let’s recall the expression we need to simplify:

[tex]\[ \frac{(4a^3 b^8 c^5 d^{-1} e f^{-4})^2 \cdot (19 a b c d^4)^0}{a^3 b^{-9} c d^{-1}} \][/tex]

### Simplifying the Numerator

1. Simplify [tex]\((19 a b c d^4)^0\)[/tex]:

Anything raised to the power of zero is [tex]\(1\)[/tex]:

[tex]\[ (19 a b c d^4)^0 = 1 \][/tex]

So, the numerator simplifies to:

[tex]\[ (4a^3 b^8 c^5 d^{-1} e f^{-4})^2 \][/tex]

2. Expand the exponent in the numerator:

[tex]\[ (4a^3 b^8 c^5 d^{-1} e f^{-4})^2 = 4^2 (a^3)^2 (b^8)^2 (c^5)^2 (d^{-1})^2 e^2 (f^{-4})^2 \][/tex]

Simplified, this becomes:

[tex]\[ 16 a^6 b^{16} c^{10} d^{-2} e^2 f^{-8} \][/tex]

### Simplifying the Denominator

The denominator remains as:

[tex]\[ a^3 b^{-9} c d^{-1} \][/tex]

### Combining the Numerator and Denominator

Now we combine the simplified numerator and denominator:

[tex]\[ \frac{16 a^6 b^{16} c^{10} d^{-2} e^2 f^{-8}}{a^3 b^{-9} c d^{-1}} \][/tex]

### Simplifying the Expression

1. Combine the exponents for variable [tex]\(a\)[/tex]:

Using the property [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]:

[tex]\[ a^{6-3} = a^3 \][/tex]

2. Combine the exponents for variable [tex]\(b\)[/tex]:

[tex]\[ b^{16-(-9)} = b^{16+9} = b^{25} \][/tex]

3. Combine the exponents for variable [tex]\(c\)[/tex]:

[tex]\[ c^{10-1} = c^9 \][/tex]

4. Combine the exponents for variable [tex]\(d\)[/tex]:

[tex]\[ d^{-2-(-1)} = d^{-2+1} = d^{-1} \][/tex]

5. Combine [tex]\(e\)[/tex] and [tex]\(f\)[/tex]:

Variables [tex]\(e\)[/tex] and [tex]\(f\)[/tex] do not have equivalents in the denominator to simplify further:

[tex]\[ e^2 \quad \text{and} \quad f^{-8} \][/tex]

So, the fully simplified expression is:

[tex]\[ 16 a^3 b^{25} c^9 d^{-1} e^2 f^{-8} \][/tex]

### Summary of Simplified Expression

The simplified form of the given expression:

[tex]\[ \boxed{16 a^3 b^{25} c^9 d^{-1} e^2 f^{-8}} \][/tex]

This completes the step-by-step simplification.
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