Simplified Expression by Dividing Numerical Coefficients and Variables with Exponents
To simplify the given expression, divide the terms in the numerator by the terms in the denominator.
The expression is:
\[ \frac{7x^3y^{-5}}{21y^6} \]
Divide the numerical coefficients:
\[ \frac{7}{21} = \frac{1}{3} \]
For the variables, use the laws of exponents to divide the terms with the same base:
\[ x^{a} \div x^{b} = x^{a-b} \]
\[ y^{a} \div y^{b} = y^{a-b} \]
Applying the laws of exponents, you get:
\[ x^3 \div x^0 = x^{3-0} = x^3 \]
(Any term to the power of 0 is 1, which is why the x term in the denominator is considered \(x^0\))
\[ y^{-5} \div y^6 = y^{-5-6} = y^{-11} \]
Now put it all together:
\[ \frac{1}{3} x^3 y^{-11} \]
The negative exponent indicates the reciprocal:
\[ y^{-11} = \frac{1}{y^{11}} \]
Thus, the simplified expression is:
\[ \frac{x^3}{3y^{11}} \]
