Identify base and exponents: Identify the base and exponents in the expression. We have the expression 3^(3)*(3^(4))^(4). Here, the base is 3, and we have two exponents, 3 and 4. The second term has an exponent of 4 outside the parentheses, which applies to the result of 3^(4) inside the parentheses.Apply power of power rule: Apply the power of a power rule to the second term. The power of a power rule states that (a^b)^c = a^(b*c). We apply this rule to the second term (3^(4))^(4). (3^(4))^(4) = 3^(4*4) = 3^(16)Combine terms using product of powers rule: Combine the terms using the product of powers rule. The product of powers rule states that a^m * a^n = a^(m+n) when the bases are the same. We combine the first term 3^(3) with the result from Step 2, which is 3^(16). 3^(3) * 3^(16) = 3^(3+16) = 3^(19)Simplify final expression: Simplify the final expression. We have simplified the expression to 3^(19). This is the final simplified form of the original expression.

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$3^{3} \cdot\left(3^{4}\right)^{4}$

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Algebra 1

Multiplication with rational exponents

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3^{3} \cdot\left(3^{4}\right)^{4}

Identify base and exponents: Identify the base and exponents in the expression. $\newline$ We have the expression $3^{3}\times(3^{4})^{4}$ . Here, the base is $3$ , and we have two exponents, $3$ and $4$ . The second term has an exponent of $4$ outside the parentheses, which applies to the result of $3^{4}$ inside the parentheses.
Apply power of power rule: Apply the power of a power rule to the second term. $\newline$ The power of a power rule states that $(a^b)^c = a^{(b*c)}$ . We apply this rule to the second term $(3^{4})^{4}$ . $\newline$ $(3^{4})^{4} = 3^{(4*4)} = 3^{16}$
Combine terms using product of powers rule: Combine the terms using the product of powers rule. $\newline$ The product of powers rule states that $a^m \times a^n = a^{m+n}$ when the bases are the same. We combine the first term $3^{3}$ with the result from Step $2$ , which is $3^{16}$ . $\newline$ $3^{3} \times 3^{16} = 3^{3+16} = 3^{19}$
Simplify final expression: Simplify the final expression. $\newline$ We have simplified the expression to $3^{19}$ . This is the final simplified form of the original expression.