Questions: The expression [ fracleft(x^4 y^3right)^4left(x y^4right)^-3x^4 y^6 ] equals fracx^ey^d where the exponent e of x is , the exponent d of y is

Solution Steps

To simplify the given expression, we need to apply the laws of exponents. First, distribute the exponents in the numerator and denominator. Then, combine the exponents of like bases by adding or subtracting them as appropriate. Finally, express the simplified form in terms of \( \frac{x^e}{y^d} \).

We start with the expression

\[ \frac{\left(x^{4} y^{3}\right)^{4}\left(x y^{4}\right)^{-3}}{x^{4} y^{6}}. \]

First, we apply the power of a power rule and the negative exponent rule:

\[ \left(x^{4} y^{3}\right)^{4} = x^{16} y^{12} \] and \[ \left(x y^{4}\right)^{-3} = x^{-3} y^{-12}. \]

Now, we combine the terms in the numerator:

\[ x^{16} y^{12} \cdot x^{-3} y^{-12} = x^{16 - 3} y^{12 - 12} = x^{13} y^{0} = x^{13}. \]

Now, we substitute back into the original expression:

\[ \frac{x^{13}}{x^{4} y^{6}} = \frac{x^{13 - 4}}{y^{6}} = \frac{x^{9}}{y^{6}}. \]

From the simplified expression \( \frac{x^{9}}{y^{6}} \), we can identify the exponents:

• The exponent \( e \) of \( x \) is \( 9 \).
• The exponent \( d \) of \( y \) is \( 6 \).

Final Answer