(12x^6y^3z^10 over 3x^4y^2z^7)^3

Question

Find the coefficient and exponent of x

Answer 1

To simplify the expression \(\left(\frac{12x^6y^3z^{10}}{3x^4y^2z^7}\right)^3\), we first simplify the fraction inside the parentheses:

1. Divide the coefficients: \[ \frac{12}{3} = 4 \]

2. Divide the powers of \(x\): \[ \frac{x^6}{x^4} = x^{6-4} = x^2 \]

3. Divide the powers of \(y\): \[ \frac{y^3}{y^2} = y^{3-2} = y^1 = y \]

4. Divide the powers of \(z\): \[ \frac{z^{10}}{z^7} = z^{10-7} = z^3 \]

So after simplifying, we have: \[ \frac{12x^6y^3z^{10}}{3x^4y^2z^7} = 4x^2y z^3 \]

Next, we raise the entire expression to the power of 3: \[ (4x^2y z^3)^3 \]

Now we apply the exponent to each part:

1. Coefficient: \[ 4^3 = 64 \]

2. Power of \(x\): \[ (x^2)^3 = x^{2 \cdot 3} = x^6 \]

3. Power of \(y\): \[ (y)^3 = y^3 \]

4. Power of \(z\): \[ (z^3)^3 = z^{3 \cdot 3} = z^9 \]

Combining all these, we get: \[ 64x^6y^3z^9 \]

Final Answer

• The coefficient is \(64\).
• The exponent of \(x\) is \(6\).