Exponents and Radicals

Using the rule of exponents, ( a^m \cdot a^n = a^(m+n) ), we have ( 3^2 \cdot 3^4 = 3^(2+4) = 3^8 ).

The square root of a number is the value that, when multiplied by itself, gives the original number. Therefore, ( \sqrt{16} = 4 ), since ( 4 \cdot 4 = 16 ).

Using the rule of exponents, ( (a^m)^n = a^(m \cdot n) ), we have ( (25^2)^3 = 25^(2 \cdot 3) = 25^6 = 5^{2 \cdot 6} = 5^{12} ).

The cube root of a number is the value that, when multiplied by itself three times, gives the original number. Therefore, ( \sqrt[3]{8} = 2 ), since ( 2 \cdot 2 \cdot 2 = 8 ).

Simplifying the expression, we have ( \sqrt{49} - \sqrt{16} = 7 - 4 = 3 ).

Using the rule of exponents, ( a^m \cdot a^n = a^(m+n) ), we have ( 2^3 \cdot 2^5 = 2^(3+5) = 2^8 ).

Simplifying the expression, we have ( (\sqrt{9})^2 = 9^2 = 81 ). Therefore, the answer is ( 9 ).

The fourth root of a number is the value that, when multiplied by itself four times, gives the original number. Therefore, ( \sqrt[4]{625} = 5 ), since ( 5 \cdot 5 \cdot 5 \cdot 5 = 625 ).

Simplifying the expression, we have ( \sqrt{100} + \sqrt{25} = 10 + 5 = 15 ).

Any number raised to the power of 0 is equal to 1. Therefore, ( 5^0 = 1 ).

Using the rule of exponents, ( a^{m/n} \cdot a^{p/n} = a^{(m+p)/n} ), we have ( \sqrt[3]{27} \cdot \sqrt[3]{9} = \sqrt[3]{27 \cdot 9} = \sqrt[3]{243} = 3 \cdot 3 \cdot 3 = 9 ).

The fifth root of a number is the value that, when multiplied by itself five times, gives the original number. Therefore, ( \sqrt[5]{32} = 2 ), since ( 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32 ).

Using the rule of exponents, ( (a^m)^n = a^(m \cdot n) ), we have ( (3^2)^3 \cdot 3^4 = 3^(2 \cdot 3) \cdot 3^4 = 3^6 \cdot 3^4 = 3^(6+4) = 3^{10} ).

Simplifying the expression, we have ( \sqrt{144} - \sqrt{36} = 12 - 6 = 6 ).

Using the rule of exponents, ( a^{m/n} \cdot a^{p/n} = a^{(m+p)/n} ), we have ( \sqrt[6]{64} \cdot \sqrt[6]{16} = \sqrt[6]{64 \cdot 16} = \sqrt[6]{1024} = 2 \cdot 2 \cdot 2 \cdot 2 = 4 ).