Pregunta
Fully simplify using only positive exponents. \[ \frac{4 x^{8} y^{5}}{4 x^{8} y^{2}} \]
Ask by Stuart Craig. in the United States
Feb 03,2025
Solución de tutoría real
Respuesta verificada por el tutor
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\( y^{3} \)
Solución
To simplify the expression \(\frac{4 x^{8} y^{5}}{4 x^{8} y^{2}}\) using only positive exponents, follow these steps:
1. **Simplify the numerical coefficients:**
\[
\frac{4}{4} = 1
\]
2. **Simplify the \(x\)-terms:**
\[
\frac{x^{8}}{x^{8}} = x^{8-8} = x^{0} = 1
\]
(Recall that any non-zero number raised to the power of 0 is 1.)
3. **Simplify the \(y\)-terms:**
\[
\frac{y^{5}}{y^{2}} = y^{5-2} = y^{3}
\]
4. **Combine the simplified parts:**
\[
1 \times 1 \times y^{3} = y^{3}
\]
**Final Answer:**
\[
y^{3}
\]
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First, notice that the coefficients of 4 in the numerator and denominator cancel each other out, giving us: \[ \frac{4 x^{8} y^{5}}{4 x^{8} y^{2}} = \frac{x^{8} y^{5}}{x^{8} y^{2}}. \] Next, we can simplify the \(x^{8}\) terms. Since the bases are the same, we use the exponent rule: \[ \frac{a^m}{a^n} = a^{m-n}. \] This gives us: \[ \frac{x^{8}}{x^{8}} = x^{8-8} = x^{0}, \] and since \(x^{0} = 1\), this term simplifies to 1. Now, let's simplify the \(y\) terms: \[ \frac{y^{5}}{y^{2}} = y^{5-2} = y^{3}. \] Therefore, the entire expression simplifies to: \[ 1 \cdot y^{3} = y^{3}. \] The final simplified result, using only positive exponents, is: \[ y^{3}. \]
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