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Quiz 14
1.) Factor completely: $3a^2b+9ab^2$.
Solution: Since $3$ is in common to both terms, $a$ is in common, and $b$ is in common, we will factor our $3ab$ to get $$3ab(a+3b).$$

2.) Simplify and write with no negative exponent: $(3a^{-2}b^2)^3$.
Solution: First we use the rule that $(xyz)^n=x^ny^nz^n$ to get $$3^3 (a^{-2})^3 (b^2)^3.$$ We simplify this using the rule that $(x^n)^m=x^{mn}$ (and for $3^3$ we simply compute $3^3=27$): $$27 a^{-6} b^6.$$ Now we make the negative exponent go away by recalling that a negative exponent means to put into the denominator of a fraction: $$\dfrac{27b^6}{a^6}.$$

3.) Write using scientific notation: $0.000000397$.
Solution: $3.97 \times 10^{-7}$

4.) Factor by grouping: $4x^3+2x^2+2x+1$.
Solution: We group the terms and factor them: $$\begin{array}{ll} 4x^3+2x^2+2x+1 &= (4x^3+2x^2) + (2x+1) \\ &= 2x^2(2x+1) + (2x+1) \\ &= (2x^2+1)(2x+1). \end{array}$$