Simplify the following expression.

[tex]\[
\frac{b^{-3} c^0 d^2}{e^{-4}}
\][/tex]

[tex]\[
\frac{d^{[?]} e^{\square}}{b}
\][/tex]



Answer :

Let's simplify the given expression step by step:
[tex]\[ \frac{b^{-3} c^0 d^2}{e^{-4}} \][/tex]

1. Consider the term [tex]\(c^0\)[/tex]:

By the properties of exponents, any number raised to the power of 0 is 1. So, [tex]\(c^0 = 1\)[/tex]. This simplifies the expression to:
[tex]\[ \frac{b^{-3} \cdot 1 \cdot d^2}{e^{-4}} = \frac{b^{-3} d^2}{e^{-4}} \][/tex]

2. Simplify [tex]\(e^{-4}\)[/tex] in the denominator:

An exponent of [tex]\(-4\)[/tex] in the denominator can be brought to the numerator by changing the sign of the exponent. This property tells us:
[tex]\[ e^{-4} = \frac{1}{e^4} \][/tex]
Therefore,
[tex]\[ \frac{b^{-3} d^2}{e^{-4}} = b^{-3} d^2 \cdot e^4 \][/tex]

3. Combine all terms in the numerator:

Rewriting the expression with all terms in the numerator, we get:
[tex]\[ b^{-3} d^2 e^4 \][/tex]

4. Eliminate negative exponent [tex]\( b^{-3} \)[/tex]:

[tex]\( b^{-3} \)[/tex] in the numerator implies:
[tex]\[ b^{-3} = \frac{1}{b^3} \][/tex]
Therefore,
[tex]\[ b^{-3} d^2 e^4 = \frac{d^2 e^4}{b^3} \][/tex]

5. Converting above step to a fraction form:

After simplifying, we see the terms inside a fraction:
[tex]\[ \frac{d^2 e^4}{b^3} \][/tex]

Hence, the final simplified expression is as follows:
[tex]\[ \frac{d^2 e^4}{b^3} \][/tex]

In the final form:
[tex]\[ \frac{d^{[2]} e^{4}}{b^3} \][/tex]
So the values are:
[tex]\[ d^{2} e^{4}, b^3 \][/tex]
which gives:
[tex]\[ (2, 4, 3) \][/tex]

Therefore, simplified result in the given expression is:
[tex]\[ \frac{d^2 e^4}{b^3} \][/tex]