To solve the given equation [tex]\( x^{12} y^8 \cdot ? = x^{20} y^{14} \)[/tex], we need to determine the exponents of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy this equation.
First, let's break down the equation step by step:
### Step 1: Isolate the unknown term
Given equation:
[tex]\[ x^{12} y^8 \cdot ? = x^{20} y^{14} \][/tex]
### Step 2: Handle the [tex]\( x \)[/tex]-terms
We need to find the exponent of [tex]\( x \)[/tex] in the unknown term. Let's denote the unknown term as [tex]\( x^a \cdot y^b \)[/tex].
Consider just the [tex]\( x \)[/tex]-terms:
[tex]\[ x^{12} \cdot x^a = x^{20} \][/tex]
Combine the exponents on the left-hand side:
[tex]\[ x^{12 + a} = x^{20} \][/tex]
Since the bases are equal, we can set the exponents equal to each other:
[tex]\[ 12 + a = 20 \][/tex]
Solve for [tex]\( a \)[/tex]:
[tex]\[ a = 20 - 12 \][/tex]
[tex]\[ a = 8 \][/tex]
So, the exponent of [tex]\( x \)[/tex] in the unknown term is 8.
### Step 3: Handle the [tex]\( y \)[/tex]-terms
Next, we need to find the exponent of [tex]\( y \)[/tex] in the unknown term.
Consider just the [tex]\( y \)[/tex]-terms:
[tex]\[ y^8 \cdot y^b = y^{14} \][/tex]
Combine the exponents on the left-hand side:
[tex]\[ y^{8 + b} = y^{14} \][/tex]
Since the bases are equal, we can set the exponents equal to each other:
[tex]\[ 8 + b = 14 \][/tex]
Solve for [tex]\( b \)[/tex]:
[tex]\[ b = 14 - 8 \][/tex]
[tex]\[ b = 6 \][/tex]
So, the exponent of [tex]\( y \)[/tex] in the unknown term is 6.
### Conclusion
The unknown term that satisfies the given equation [tex]\( x^{12} y^8 \cdot ? = x^{20} y^{14} \)[/tex] is:
[tex]\[ x^8 y^6 \][/tex]
In summary, the exponents of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] in the unknown term are 8 and 6 respectively.
