To simplify the expression [tex]\(\left(3 x^3 y\right)^{-1} \times\left(6 x^5 y^7\right)\)[/tex], we will apply the laws of indices step by step. Here's the detailed process:
1. Express the given term separately:
[tex]\[
\left(3 x^3 y\right)^{-1} \times (6 x^5 y^7)
\][/tex]
2. Apply the negative exponent rule:
The negative exponent means taking the reciprocal of the term. Hence, [tex]\(\left(3 x^3 y\right)^{-1}\)[/tex] becomes:
[tex]\[
\left(3 x^3 y\right)^{-1} = \frac{1}{3 x^3 y}
\][/tex]
Now we have:
[tex]\[
\frac{1}{3 x^3 y} \times (6 x^5 y^7)
\][/tex]
3. Combine the fractions:
[tex]\[
\frac{6 x^5 y^7}{3 x^3 y}
\][/tex]
4. Simplify the constants:
Simplify [tex]\(\frac{6}{3}\)[/tex]:
[tex]\[
\frac{6}{3} = 2
\][/tex]
Now we have:
[tex]\[
2 \times \frac{x^5 y^7}{x^3 y}
\][/tex]
5. Apply the laws of exponents to the variables:
- For [tex]\(x\)[/tex], use the rule [tex]\(\frac{x^a}{x^b} = x^{a-b}\)[/tex]:
[tex]\[
\frac{x^5}{x^3} = x^{5-3} = x^2
\][/tex]
- For [tex]\(y\)[/tex], use the rule [tex]\(\frac{y^m}{y^n} = y^{m-n}\)[/tex]:
[tex]\[
\frac{y^7}{y} = y^{7-1} = y^6
\][/tex]
6. Combine the simplified terms:
[tex]\[
2 \times x^2 \times y^6
\][/tex]
Thus, the simplified expression is:
[tex]\[
2 x^2 y^6
\][/tex]
