To simplify the given expression [tex]\((-3)^3(-3)^7 y^{10}\)[/tex], we can use the properties of exponents. Here's a detailed, step-by-step solution:
1. Identify the common bases: We see that [tex]\( (-3)^3 \)[/tex] and [tex]\( (-3)^7 \)[/tex] both have the same base, which is [tex]\(-3\)[/tex].
2. Apply the properties of exponents: Recall the property of exponents that states [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]. This means we can combine the exponents when multiplying terms with the same base.
[tex]\[
(-3)^3(-3)^7 = (-3)^{3+7}
\][/tex]
3. Add the exponents: Solve the exponent addition inside the parentheses.
[tex]\[
(-3)^{3+7} = (-3)^{10}
\][/tex]
4. Simplified base term: Now, we have the base term [tex]\((-3)^{10}\)[/tex].
5. Preserve the second term: The expression [tex]\(y^{10}\)[/tex] remains as it is, since it does not need any further simplification.
6. Combine the results to form the final expression: Now we have simplified the given product to two separate terms.
Thus, the simplified expression is:
[tex]\[
(-3)^{10} y^{10}
\][/tex]
Given the information and the numerical result from earlier steps:
[tex]\[
(-3)^{10} = 59049
\][/tex]
So, the fully simplified expression with the exact numerical value is:
[tex]\[
59049 y^{10}
\][/tex]
Therefore, [tex]\((-3)^3(-3)^7 y^{10}\)[/tex] simplifies to:
[tex]\[
59049 y^{10}
\][/tex]
