Understand the expression: Understand the expression 9^(log_(3)20). We have an exponentiation where the base is 9 and the exponent is the logarithm of 20 with base 3.Express base in terms: Express the base 9 in terms of base 3. Since 9 is 3 squared (3^2), we can rewrite the base as 3^2.Substitute base in expression: Substitute the expression for 9 into the original expression. Now we have (3^2)^(log_(3)20).Apply power rule for exponents: Apply the power rule for exponents (a^(m*n) = (a^m)^n). We can rewrite the expression as 3^(2*log_(3)20).Use logarithm power rule: Use the logarithm power rule. The power rule states that a*log_b(c) = log_b(c^a). Therefore, we can rewrite the expression as 3^(log_(3)(20^2)).Simplify using property of logarithms: Simplify the expression using the property of logarithms that b^(log_b(x)) = x. Since the base of the logarithm and the base of the exponentiation are the same (3), the expression simplifies to just 20^2.Calculate 20^2: Calculate 20^2. 20^2 is 400.

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Math Problems

Algebra 1

Evaluate integers raised to positive rational exponents

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9^{\log_{3}20}

Understand the expression: Understand the expression $9^{\log_{3}20}$ . We have an exponentiation where the base is $9$ and the exponent is the logarithm of $20$ with base $3$ .
Express base in terms: Express the base $9$ in terms of base $3$ . $\newline$ Since $9$ is $3$ squared ( $3^2$ ), we can rewrite the base as $3^2$ .
Substitute base in expression: Substitute the expression for $9$ into the original expression. $\newline$ Now we have $(3^2)^{(\log_{3}20)}$ .
Apply power rule for exponents: Apply the power rule for exponents $a^{m*n} = (a^m)^n$ . We can rewrite the expression as $3^{2*\log_{3}20}$ .
Use logarithm power rule: Use the logarithm power rule. $\newline$ The power rule states that $a\log_b(c) = \log_b(c^a)$ . Therefore, we can rewrite the expression as $3^{\log_{3}(20^2)}$ .
Simplify using property of logarithms: Simplify the expression using the property of logarithms that $b^{\log_b(x)} = x$ . Since the base of the logarithm and the base of the exponentiation are the same $(3)$ , the expression simplifies to just $20^2$ .
Calculate $20^2$ : Calculate $20^2$ . $\newline$ $20^2$ is $400$ .