Q. \[ \frac{32x^3y^2z^5}{-8xyz^2} \]

Q: What is the simplified form of \frac{32x^3y^2z^5}{-8xyz^2} ?

Simplify coefficients and subtract exponents: 32/ -8 = -4, x^{3-1}=x^2, y^{2-1}=y, z^{5-2}=z^3. Final: -4x^2yz^3 .

Q: How do you divide the numerical coefficients 32 and -8?

Divide normally: 32 \div -8 = -4. The negative sign stays since one factor is negative.

Q: Why do you subtract exponents when dividing like bases?

Division of powers uses a^m/a^n = a^{m-n}. You subtract exponents because division cancels common factors of the base.

Q: How do you treat variables written without an exponent, like x or z?

An unwritten exponent means 1: x = x^1. Use that when subtracting exponents (e.g., x^3/x = x^{3-1} = x^2).

Answer

Divide coefficients and subtract exponents:
\[
\frac{32x^{3}y^{2}z^{5}}{-8xyz^{2}}=\frac{32}{-8}x^{3-1}y^{2-1}z^{5-2}=-4x^{2}yz^{3}.
\]

Detailed Explanation

We simplify the expression step by step:

Write the original expression.

\[
\frac{32x^{3}y^{2}z^{5}}{-8xyz^{2}}
\]
Simplify the numerical coefficients separately: divide 32 by -8.

\[
\frac{32}{-8} = -4
\]
Simplify the powers of x by subtracting exponents (division law: a^{m}/a^{n} = a^{m-n}). For x: subtract 1 from 3.

\[
\frac{x^{3}}{x} = x^{3-1} = x^{2}
\]
Simplify the powers of y: subtract 1 from 2.

\[
\frac{y^{2}}{y} = y^{2-1} = y^{1} = y
\]
Simplify the powers of z: subtract 2 from 5.

\[
\frac{z^{5}}{z^{2}} = z^{5-2} = z^{3}
\]
Combine the simplified parts (numerical coefficient and variables).

\[
-4 \cdot x^{2} \cdot y \cdot z^{3}
\]

Final simplified result:

\[
-4x^{2}y z^{3}
\]

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FAQs

What is the simplified form of \( \frac{32x^3y^2z^5}{-8xyz^2} \)?

Simplify coefficients and subtract exponents: \(32/ -8 = -4\), \(x^{3-1}=x^2\), \(y^{2-1}=y\), \(z^{5-2}=z^3\). Final: \( -4x^2yz^3 \).

How do you divide the numerical coefficients \(32\) and \(-8\)?

Divide normally: \(32 \div -8 = -4\). The negative sign stays since one factor is negative.

Why do you subtract exponents when dividing like bases?

Division of powers uses \(a^m/a^n = a^{m-n}\). You subtract exponents because division cancels common factors of the base.

What if a variable appears only in the numerator or only in the denominator?

If only in numerator, it remains with its exponent. If only in denominator, subtracting exponents gives a negative exponent; you can rewrite as a reciprocal if needed (e.g., \(a^{-1}=1/a\)).

How do you treat variables written without an exponent, like \(x\) or \(z\)?

An unwritten exponent means 1: \(x = x^1\). Use that when subtracting exponents (e.g., \(x^3/x = x^{3-1} = x^2\)).

Can you cancel common factors before multiplying out?

What if the exponent subtraction gives zero, e.g., \(x^2/x^2\)?

Any base to the zero power equals 1 (provided base ≠ 0). So \(x^2/x^2 = x^{0} = 1\), and the variable disappears from the simplified result.

Are there domain restrictions (values you cannot use) for \( \frac{32x^3y^2z^5}{-8xyz^2} \)?

Yes: you cannot have any variable in the original denominator equal to zero, so \(x \neq 0\), \(y \neq 0\), and \(z \neq 0\); otherwise the expression is undefined.

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